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Essex phd thesis

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How To Create A Modern CV/Resume With InDesign. In this tutorial we’re going to be learning how to use InDesign to create a clean and structured CV/Resume design. Essex. We will be using basic tools and techniques (but ever so important ones!) to create the design. If you haven’t already, you should check out our “Getting To Grips With InDesign” series to jerry hawkins brush up on essex phd thesis, your InDesign skills. As with all designs, we need something to refer to when designing. Grab a piece of paper and sketch out a couple of ideas for your CV’s structure. The CV I’m designing is going to block cover letter be double-sided – yes, it costs a little more to essex produce, but it gives you a little bit of space to showcase some of your work and ultimately “wow’s” your hopefully future employer even more! It pays to get resumes professionally printed – if it lands you a well-paid job that you love it is well worth it! Sketches don’t have to take long – I spent just a couple of minutes on mine. It just helps avoid that “ what do I do? ” phrase when you open up a blank screen in InDesign or any other application!

It’s now time to open up InDesign and jerry michael hawkins dissertation set up a document. When InDesign is open, go to File New Document . Select A4 as your page size, and change the number of pages to two, making sure Facing Pages is unchecked. Change your margins down to 5mm and your bleed up to 3mm . Click OK. You’ll see that we have three different lines, a red one (which is our bleed line – any artwork that meets the edge of the phd thesis page should meet this red line), the term papers on ups black line (this is our actual page) and our purple/pink line which is our margin line (all content such as text etc should be inside this line). The next step is to start putting together some of our structure, which is phd thesis possibly the hardest thing to write a descriptive essay do as we don’t actually know how much space we need for essex phd thesis, our different sections until it’s all written up. However, it’s still a good idea to get some of the structure sorted. Defense. Grab the Text Tool and essex drag out a selection in the top left corner of your page. This will be our text box for our “ CV/Resume ” title. You’ll notice when you drag out a new text box it will automatically snap to the alignment of thesis defense other objects and/or important areas, such as the essex phd thesis center of the page.

Drag out another text box for your first header and your basic information section, such as your name and contact details. Keep on block cover letter, dragging out essex phd thesis text boxes, making sure they’re well aligned. It’s now time to do the same but with image placeholders rather than text boxes. Select the jerry michael hawkins Rectangle Frame Tool , and repeat the steps again in places we want images. I’m going to have a profile image next to essex phd thesis my profile text, and some icon images in the bottom left hand corner for write essay of packingtown, my skills. Repeat the same steps again on our second page, where we will display some of essex phd thesis our work. It’s time to start filling in our content! Lets start with out main header, “ CV/Resume “. Double-click in thesis header box and type in your words. Select a suitable typeface of your choice – I’m using Blue Highway Bold as it is the font used in my logo, and I want to keep my identity going throughout my brand. Go through your two pages filling in essex phd thesis all the header titles. Block. For this I have again used the typeface Blue Highway, set to 21pt.

Start filling in your profile section, including your name, email address, contact number, postal address and anything else you feel is necessary, such as your websites URL. I used Arial at essex phd thesis 12pt for this section, and will do for all other sections of my text. Select all of the text you just inserted and change the Leading to 16pt – the leading is the space in between each line of text. This makes it much easier to read. Insert the rest of your profile information – this should include information such as how old you are, your ambitions, goals, and hobbies.

To add some “ oomph ” to our profile, we’re going to make certain words bold and italic. Go through your profile text and change some words to Arial Bold or Italic. I also lowered the size of the typeface to write essay 11pt and the Leading to 14pt. To finish off our profile section we’re going to add a portrait picture of yourself. For this tutorial, I’m going to use a photo of myself. Click on your image placeholder in your profile section and essex then go to write essay File Place . Select your image and click OK. The chances are the image you have inserted is phd thesis a lot bigger than your image placeholder. On Ups. No worries though, this is phd thesis super easy to fix. Select your Direct Selection Tool and click on the image you have inserted into your placeholder. You will notice that this has selected the image inside the placeholder rather than the placeholder itself.

Still with the Direct Selection Tool selected, whilst holding the term on ups Shift-Key to keep the image in proportion, scale the essex phd thesis image down. If you’re image seems blurry or pixelated at all, this is most probably caused by InDesign’s default display performance, which is paper mixed between speed and quality. For a small document like this with barely any images, you’re best off using the highest quality setting. You can change this by going to View Display Performance High Quality Display . Finish the rest of the text-based tasks off using techniques we have already used. Areas we need to fill include the qualifications, education/employment and achievements areas. As you can see I have used the phd thesis same typeface through all of the text, with bold titles and italic used for a descriptive essay, things such as the dates at essex phd thesis the end of book school each achievement. Fill in the final piece of text on the back of phd thesis your CV/Resume and then we can move on! Our next step is to find a good set of icons, for defense, which we’re going to essex phd thesis be using in our skills section of our CV. We have space for 8 icons – we could of course design our own, but for of packingtown, this tutorial I’m going to phd thesis outsource to ‘freebies’ to block cover save us some time. Essex Phd Thesis. Decide which 8 skills you’d like to include. In my case, I’m going to use: OS X, Windows, Email (Communication), Photoshop, Illustrator, InDesign, TextWrangler and Microsoft Office.

Below is defense a list of where I got my icons… With that done, open all of your icons up in Photoshop and resize them to phd thesis exactly 100?100 pixels. Once done, save all of your files as JPGs. With your icons done, start placing them into your icon/image placeholders that we created earlier. Letter. You can do this the same way we inserted our profile picture; by clicking on essex phd thesis, the placeholder and then going to File Place , selecting your file and then resizing it within the image placeholder using the Direct Selection Tool . Our next step is to a descriptive essay make some blocks for our ‘meter readings’ to show our skill level of the different applications. Phd Thesis. Select the Rectangle Tool and drag out a shape with the block cover same height as your icons. Copy and paste the shape several times, and set them an equal space apart from each other as seen below. Remove the stroke of all the shapes, and color them all grey. Copy and paste all of the phd thesis shapes seven times, placing them in the correct position next to your other icons. Fill in your skill level for editing english, each application – be honest here, there’s no point in phd thesis lying, they will find you out! The next stage is to add your portfolio of work into editing english the image placeholders on our back page.

To do this, select the essex phd thesis image placeholder and cover the go to File Place to locate your image. Resize the placeholders using the essex Selection Tool, and the placeholders contents using the term papers on ups Direct Selection Tool . With that done, so is our CV/Resume! The next step is to export our document as a PDF for easy printing and phd thesis sending digitally via email. Having a PDF version of dissertation your CV is also great for showing it off on shiny gadgets such as your iPad! Go to File Export and save as a PDF . Select “ Smallest File Size ” from the essex phd thesis PDF settings. As our CV is made up of just text and few images, the smallest file size quality should be fine for both on-screen and sending. Editing English. The biggest bonus is that it should end up being a tiny file to send over email. And that’s it, we’re done!

Just remember that it’s worth spending a little bit of quality time when putting your CV together; it should not only show off your creative ideas and skills, but also give away a clue or two into your personality and phd thesis the person you are, leaving the viewer wanting to find out more about you, hopefully leading to an interview, or at least a phone call! We’d love to see your CV and Resume designs, so why not share a link to your work in the comments? Callum Chapman is a self-employed designer, illustrator blogger. View his work at block cover Circlebox Creative. Essex Phd Thesis. He is also the founder of jerry michael Picmix Store, a store dedicated to limited edition prints, and The Inspiration Blog. Essex. Drop him a line on Twitter! It’s a really good model for resume! Very creative! ;) Craig Harrison says.

It looks quite cheap and quick. If someone came to me with that at my company, I wouldn’t have great first impressions with their design capabilities. yo Craig, easy man… a resumes not a piece of design work :-) its just a list to give you an overview of what a person has done before . could be nicely edited indeed. actually this list of term on ups tools you are working with is essex phd thesis quite obsolet… I dont know a professional who doesnt expect photoshop experience from jerry hawkins dissertation, a graphic designer as well as nobody is interested in essex phd thesis what editor you are coding as long as it is NOT dreamweaver. and as you wont apply for a job as a secretary, who cares how deep you are into something like MS office Word :-) I second craig actually. Doesn’t feel special in any way, quite texty if you ask me. @Dennis : The icons are there as an example (and by the way, Photoshop is in there). Jerry Michael Hawkins Dissertation. The goal with this tutorial wasn’t to actually design the most incredible resume ever, it was to essex phd thesis show techniques that can be used to create your own. Grab the InDesign file from the defense tutorial and essex use it as a starting point :) Its a great tutorial and I might use it for myself.

However you can not use a photo of yourself on the resume. Most places will throw it out for legal reasons, they’re terrified that someone will complain they weren’t hired because of how they looked or the color of their skin or because they have a minaret in papers on ups the background ;) This is an awesome tutorial! I love that you sketched first. Essex Phd Thesis. I always do that too. Jerry. I love how you defined your skills. That is always the hardest part for me to essex phd thesis put into jerry dissertation words.

If you use InDesign to make your resume, use the advanced tools to make a baseline grid, at least. To make a resume like the essex one you made, Illustrator is enough, and easier. Would you really show your skill level that way? Looks like an easy way to show what your not good at. About half the content on that CV/Resume shouldn’t even be there. Before you even start the layout, sort out the content. Function before form. @Illet : Good point, but on the other hand it’s also a way to tell clients/employers what you are good at. If you ‘can’ work with a particular application but you’re not at the advanced/expert level yet, it’s still worth mentioning in my opinion.

@Lennington : What would you add/remove then? The goal with this tutorial was to show how you can create your own resume with InDesign (a starting point), not to creative high tell people what should or should not go on there. You will of course put different content depending on essex phd thesis, the job you’re after, the industry, the income level, the position in a company, etc… Note: Do NOT ad a picture of yourself to a CV unless you want the person to judge you before meeting you. Book Reports. The hoodie etc would turn me off strait away just based on phd thesis, my personal judgment. I can say I will not judge these people but it is human nature. A CV is you on paper, not you in person. @steve: This post is not about what you should or should not put on your CV, that’s up to the person looking for papers on ups, a job (as I’ve mentioned in previous comments) and in some cases having your picture on there can help. I’ve put a picture of myself on my CVs and that never prevented me from getting a job or at least an interview. Essex Phd Thesis. If someone judges me from my looks before meeting me, I honestly don’t wanna work for them. It’s a nice way to design CV.

N seriously very nice tutorial. You have explained it very well. Christopher Anderton says. About not putting your picture in your CV. Sounds like a U.S. only thing (that part about being accused for rasism). Where i live (Sweden) you should almost ALWAYS put your picture in your CV. Great article Callum – thank you ! The CV it self isn’t really impressive in letter anyway, However! I agree with some people said, that it’s an inspiration. Daniel Winnard says. Whats your issue with Dreamweaver? I actually love seeing the mixed comments here!

Whether you like a CV/Resume, at essex the end of the day, comes down to personal taste. I’ve seen CV’s I’ve loved and CV’s I’ve hated, yet those with CV’s I didn’t like have great jobs, and those with CV’s I do like are unemployed… It totally depends on who is looking at creative book reports high the CV. @Steve: A lot of CV’s have pictures on… If I were reading through CV’s I personally would prefer those with photos of themselves. I think it shows they have the confidence to think outside of the phd thesis box, try new things and papers on ups take little risks. That’s a good thing, especially in the design industry. And that picture is just an example of me – doesn’t mean you HAVE to have a hood up, and besides, what’s wrong with a hood? You can’t be stereotypical when it comes to essex phd thesis CV, everyone should have an write, equal right when it comes to appearance etc. As Jon said, if I didn’t get an interview/job because of my appearance I wouldn’t want to work for them anyway!

@Illet: It’s just as important to show the things you’re not great at essex as it is the things you are great at. It a) shows you’re honest, b) know you have room to michael hawkins improve with certain skills and applications and c) shows you want to improve – otherwise you wouldn’t have put it there. Employers like to know you want to learn and want to essex keep up with the fast-moving trends, regularly updated software, development skills and thesis defense last but not least, technology. @Dennis: It doesn’t just indicate how good you are at the application you develop in, but how confident you at phd thesis developing all together. You wouldn’t give yourself 4/5 for a coding application if you couldn’t code… But you could use a HTML icon or something! Also, lots of term employers like to know they can ask you to essex help out with letters etc. Term Papers On Ups. There’s no harm in essex phd thesis letting people know you have good office skills, too. :)

It’s always nice to book high school see examples of different and customized CVs… Really creative and nice CV! thanks for phd thesis, tutorial! I adopted inspiration and changed my own resume. Tips are always welcome :) Geez.. the title of the creative book reports post was how to CREATE your CV / Resume. Just to show you how to lay the foundation. It wasn’t meant to be the be all end all on how to DESIGN one. @Jon It’s all about either being personal or being TOO personal. Let’s say that the essex company you applied for is affirmative action company, including your picture can tip the creative reports high scales either for or against you. Designers can be self-righteous all they want but business is phd thesis business. Believe or not if you put your age on your resume, the employer has the term on ups ability to discriminate you without breaking any laws by asking for your age.

This is a HUGE no. People need to know their rights, like what are “illegal” questions an employer can’t ask. Plus, putting images of phd thesis yourself leaves you wide open. Designer love to be personal and creative but giving alway too much personal information can hurt you in the long run. Editing English. That’s my experience in essex the industry in the US. Like Christopher said, it differs from country to write a descriptive essay country. In my opinion, there is nothing wrong approaching a potential employer with a personal approach but sharing your age and including a photo can be hurtful to your price quote (younger = cheaper + easier to get free service for example). Very good tutorial for example. @Patricia : Thanks for your comment, you bring good points. In my opinion, if someone discriminates me because of my age or what I look like, like I said, I wouldn’t want to work for essex phd thesis, them anyway, but on the other hand I agree that having your age and a picture of yourself on cover letter, your CV may, in some cases it, hurt more than it can help. If you include your age on phd thesis, your resume, I see this as a way to weed out defense paper employers you probably wouldn’t really want to work for in the first place.

But like you said, it seems to differ from country to country. different and professional design. thanks. @Tycho: Nice work! I love the graph to show off your skills! @Patricia: Good points; I think it totally depends on where you live, what position you’re applying for, and what company you’re sending it into. Some companies are quite old fashioned and do take age, experience etc too seriously, where as there are a lot of young, creative and essex experimental agencies out there who purposely go for younger designers to bring fresh talent, modern ideas etc, in which case a picture and your age could be beneficial. Completely depends on your circumstances I guess! Thanks for your tutorial. Book Reports High School. It’s Awesome! About InDesign I have a Templates for InDesign. For example: Newsletter Templates, Book Templates and essex phd thesis much more but It’s free.

Personally I wouldnt add a photo to a Resume but thats just the done thing here in Ireland. Editing English. I know that other European countries such as Spain where employers expect to essex phd thesis see a photo on it. Regarding the tutorial (not the write essay content, cos thats not what is phd thesis being taught here!), its very well written, easy to follow and helpful to those who want to get into InDesign. Thanks for posting! I’ve come across conflicting information about whether or not you should save your resume in PDF. Creative Reports. I’ve read several design blogs that say your resume should show off your design abilities. Therefore design it in essex phd thesis InDesign and save to PDF. Write A Descriptive Essay. But some human resources professionals say that you should NEVER submit a resume in anything but MS Word format.

Anyone care to comment? I know I would be impressed if an applicant brought in essex a resume that looked that nice. Indesign is very user friendly, but going into it with a guide like this makes the job a lot easier to understand. I have heard that too, but funny when you said that, I cringed, thinking of all of the people who have brought me flattened pdfs and asked for them in a descriptive essay ms word format. I say go for indesign and convert to pdf.. why the phd thesis hell would they want to be able to edit your resume anyway! :) I love the tutorial and the comments. The tutorial IS just a tutorial on layout, hence the use of InDesign. As for the content, not every employer is the same. Nor are the industries. To be honnest, I didn’t realy like it at first. However I think this tutorial is more intended for absolute newbies in block Indesign, or people outside of the essex phd thesis design industry.

Theirfore : thumbs up ! Spot on with this article, i although have the book reports high problem that i am not a gfx wizkid as you are, so i would rather have a system that does these things for me. Phd Thesis. I always found it very time consuming to maintain different CV for different positions i wanted to apply for. Normally i make a special CV for thesis, my jobs, if they are asking for something specific in the job ad then I write in my CV the things that I have had experience with. For a long time i did this manually, but now i found a tool on the internet where you can have your CV and work with them actively. It can even inherit data from other CV’s in a way that i only have to change on place end the basic stuff gets changed automatically. For mee as a computer specialist but not a graphics wizard like you, this is an easy and transparent way to have several CV’s. It sounds like you might not be in the US, but in phd thesis the US I’m pretty sure it’s illegal to put your photo or for papers, an employer to ask for your photo. Your clients list should be much easier to phd thesis scan, why not a 3-column list or something. I hate it when design jobs ask for designers to submit their resumes in Word, hello? do you understand what design is, it’s infuriating. I did recreate my creative resume in editing english Word to satisfy those idiots (though normally I ignore them because they obviously don’t know what they’re doing.)

It looks like you used all one typeface (Arial?) and you really should try to create as much visual hierarchy as possible, which includes using different typefaces. Listing all your school grades makes it seem like you’re a high schooler trying to get a job and that you don’t know what else to phd thesis list. Qualifications to editing english me means, handled X project X under budget with X people working with/for me, something like that. A “C” in Science doesn’t show me anything except that you don’t know Science. To be honest, I wouldn’t even consider you for a design job.

The designer’s job is to essex phd thesis show people what’s important and to get them to move over the information and guide them through the page/item, but everything’s the same volume on your page. Creative High School. I wouldn’t feel confident you could get me the results I wanted. Why not step up and show us your own resume instead of leaving an anonymous comment? Another great article. I like how you showed off the draft/prototype that was originally hand drawn.

That phase… is a must when designing. I appreciate this post as it gives me a few ideas, but probably more for my Web site than for a print resume. (And this is a resume. A CV is much longer and contains citations of journal publications and the like. Essex Phd Thesis. CVs are mostly for a descriptive, academics and essex researchers applying for grants and other academic positions. It’s not really the same thing.) I especially like seeing your overall design process as that can be applied to any design project. It’s rare to creative book reports school get that glimpse into someone’s head. I can see insight and phd thesis practicality. It’s certainly not illegal to editing english put a photo of yourself on your resume, unlike “Designer’s” comment. (It’s also not illegal to essex phd thesis swear on a resume, but should you?) It’s just not always wise to show your visage though unless you’re an actor, model, etc.

The reason is, someone might assume you’re too old, too young, too white, too dark, too pretty, too ugly, etc. Believe it or not, some people won’t hire attractive men or women. So, even if you’re a god(dess), it could be a bad idea. But if that’s expected in your area of the world, then just make sure you look great and professional. I’m not sure about the printing on term papers, both sides though. I’ve had a lot of essex HR and creative directors tell me not to do that since it makes it too hard to jerry dissertation make notes and draw correlations. Plus, unless you have a great paper, it can just look cheap. And glossy paper seems to be out as it’s again too large to make notes on the resume itself. Again, thanks for the post — and the icon links.

Casanova Frankenstein says. It’s not illegal to put your photo on your CV for first amendment reasons, but it’s not commonly done for reasons of good taste. You’re right that it is illegal for the employer to ask for essex phd thesis, your photo. Great tips to creative book reports high school give someone has done a resume in awhile. Really catches your eye and well organize, and attention to detail. Erica Cain says.

I am a professional corporate recruiter (read: I work in-house for one company and am not what is commonly called a headhunter), and I can tell you that I have never heard anything more proposterous than someone saying it is phd thesis “illegal” to put anything on a resume. *facepalm* That being said, HR professionals do not want you to put a photo on your resumebecause it sets them up for the exact issues that Stanford mentioned. Depending on the field, it also can come across as quite unprofessional. A good HR professional really don’t care what their IT/marketing/communications/etc. professionals look like. If they do, then they are in the wrong field. Also, as was also mentioned by Stanford, this is cover letter not a CV, and you do people a disservice by essex labeling it as such. Some people may be searching for how to create a CV, and I’m sure you would hate for book high school, them to be misinformed. Phd Thesis. This is a resume. Get the accents right (notice there are two) because spelling is a huge deal on resumes. If you don’t have attention to thesis detail when selling yourself, then why would I think you have attention to phd thesis detail when working on thesis defense paper, my product? I agree that the qualifications section is not what I would consider a qualifications list.

Your grades don’t matter to me. I want to know if you graduated, and that’s really about it because that is an employer set requirement. That doesn’t tell me anything about phd thesis you, though. Thesis Paper. I want to know WHY to phd thesis hire you. What have you contributed in the past?

Did you implement something nifty that created an annual cost savings of 20%? Tell me about THAT! Those are the things that sell you, and that I can sell to the company. Don’t go into anymore depth than what I listed in my example. Remember–your initial goal is to jerry land an interview. You can tell me the rest then. You WANT me to want to know more about phd thesis you and editing english your projects.

Speaking of selling yourself, “One of my…” is not the way to start a bullet point that will get my attention. You should always start a line with an action verb: “Developed award-winning marketing campaign for , which led to a 10% increase in sales” is much more impressive. Don’t bore me or I will move right on to the next resume. You have about 5 seconds to win me over so that I will finish reading your resume. Phd Thesis. That is the harsh reality in editing english a world where everyone is seeking a job opportunity and HR professionals are flooded with applicants. All that being said, I love the essex icons that you use to editing english list skills. “Skills” is usually the most boring field on a resume because people write the same crap (and, yes, you are 100% right that we WILL find out if you lie on these things–often before you get your foot in the door since we often use assessments to essex phd thesis weed out the people who do lie on their resumes). This is a unique way to display skills, and it would definitely catch my eye. I know I listed a lot of constructive criticism in this post, but I do admire your goal to create an interesting and cover letter different resume. Phd Thesis. For creative job opportunities, this is high very important.

Kudos to you! I’m embarrassed at some of the comments here. Essex Phd Thesis. I love this tutorial and I think like some people have said that the point was to show how to do this in InDesign, not to tell people what to put on their resume. For me, this was very helpful. Some people need to relax.

If you have so much to say, make your own tutorial for people to rip apart. Thanks for the tips!

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How to Write an Achievement Oriented Resume. Many people run into essex phd thesis trouble when writing the details of the work experience section on jerry dissertation a resume. Commonly, the work experience section is made up of a bullet point list of essex duties and responsibilities relating to creative reports school each work position. However, in order for essex phd thesis, your resume to stand out, the details of your work experience section should ideally start with a powerful action verb, as well as using numbers to quantify your accomplishments . When writing the defense paper, work experience, always begin your bullet point details with an action verb . A powerful action verb places you as an initiator of action, which leaves a positive impression on the reader. Essex Phd Thesis? Rather than beginning a description with a passive-sounding description such as “Worked on editing english creative projects to teach children,” it is better to start off using an phd thesis, action verb such as “Designed and implemented a creative arts curriculum for elementary school children.” Try to essay of packingtown avoid starting off descriptions with “Responsible for” and instead, use action verbs such as “managed,” “implemented,” or “developed.”

For a complete list of action verbs (as in over 1,000 words) view the “Longest Action Verb List In The Universe” or get your action verbs by skill: There’s a simple formula that any job seeker can follow to construct accomplishment-oriented bullet points. It’s called the ‘PAR’ Method, which stands for problem, action, and results. When applied to your resume, the ‘Par Method’ encourages you to: Problem: Identify a responsibility or issue at phd thesis work Action: Discuss how you addressed the editing english, problem Results: What was the outcome of that action. While that may sound like a lot to fit into one bullet point, you’ll be surprised out how easy ‘PAR’ can be implemented into your bullet points. Check out the essex phd thesis, examples below: Developed new filing and organizational practices , saving the company $3,000 per year in contracted labor expenses.

Suggested a new tactic to jerry michael dissertation persuade canceling customers to essex phd thesis stay with the high school, company, resulting in a 5% decrease in cancellations. Notice that the problem, action, and result does not always need to be placed in the same order. Now that you have a better understanding of the structure of an accomplishment bullet point, let’s discuss how you can apply it to your own professional experience section. Employers want to see workers who can achieve solid results , and results are best stated in phd thesis, terms of reportable numbers. How many employees did you work with or oversee? By what percentage did you increase sales or efficiency? How much of block cover a budget did you work with, with what type of results? Putting a number on your accomplishments is a sure way of essex conveying results and impressing the hiring manager. Hiring managers like to see quantifiable achievements rather than a list of general descriptions of job responsibilities.

By using numbers in a descriptive essay, detailing your work experience, you are demonstrating your focus as being results-oriented rather than task-oriented . For example, compare “Responsible for selling products to customers at XYZ Store” to “Increased sales revenue by 30% in three months.” Which one sounds better? By including a percentage as well as time spent, the potential employer has a measurable, defined idea of what you have accomplished, rather than just a general job responsibility that can already be assumed with the job title. In order to measure your accomplishments, try to phd thesis obtain as much data as you can in defense paper, regard to your previous work experience. Essex Phd Thesis? It is never recommended to make up numbers, as hiring managers are experienced when it comes to scanning resumes and it could hurt you later on. You also do not need to thesis defense paper quantify every single line in your work experience , but at least have a few per position on phd thesis the work experience section. Below are some questions that may help to think of how to quantify achievements (broken down in terms of percentages, numbers, dollar amounts, and time) Questions to book reports high school ask yourself: Did you increase sales, market share, or customer satisfaction by a certain percentage? How? Did you increase efficiency or productivity by a certain percentage?

Did you recruit, work with, or manage a certain number of employees or teams? How many customers did you serve on average? Did you increase the number of customers served? By how much? Did you implement new ideas, systems, or processes to the company? What was the impact?

Did you propose or work with a budget of a certain dollar amount? Did you increase sales or profitability by a certain dollar amount? How? Did you decrease delivery or turnaround time on a project? How? Was one of your achievements completed within a tight deadline? Did you resolve any particular issues? How soon? All of these are examples where you can specifically quantify an achievement and translate your work experience into a results-oriented approach. In order to provide even more detail, consider also answering “How?” in essex phd thesis, regard to how you achieved the book school, accomplishment.

4. Resume achievement examples by industry. Memorized restaurant’s wine stock and phd thesis the meals they should accompany, leading to daily wine sales averaging $150 , fully 20% higher than company average Write patrons’ food orders on term slips, memorize orders, or enter orders into computers for essex phd thesis, transmittal to kitchen staff in a 110+ seat restaurant. Administrative Assistant/Office Worker. Developed new filing and organizational practices, saving the company $3,000 per year in contracted labor expenses Answered incoming calls (avg. 40/day) resolving issues with both customers and billing department. Provide direct quality care to patients including daily monitoring, recording, and evaluating of medical conditions of up to 20 patients per day Led and mentored 10 newly licensed nurses in developing and on ups achieving professional expertise. Increased students’ scores in standardized tests by essex 24% in literacy and 35% in math Educated an thesis defense paper, average of 18 students in grades 2 and 3 , and received four “Best Teacher Award”

Manage a $350,000 budget, with a reduction of costs totaling 15% over two years Trained and supervised 2 new employees, ensuring they maintain fastidious attention to detail. Consolidated multiple ticketing systems, improving communication and ticket turnover rate by essex phd thesis 7% Refined and improved existing documentation system, resulting in reduced labor costs totaling $15,000 annually via increased workplace efficiency. Operate POS cash register, handling 92 transactions on average daily, and count money in jerry michael hawkins, cash drawers to ensure the amount is correct Assist an essex phd thesis, average of 40 customers per term on ups day in finding or selecting items, and provided recommendations that generated $8K in additional revenue. Writing an achievement oriented resume is easy with our powerful and simple to use Resume Genius’ resume building software. Phd Thesis? Our bullet points are well-written and write a descriptive essay of packingtown can be easily modified to essex phd thesis reflect your achievements. Just add numbers, and you’re all done! Or, if you’d like to write your resume yourself, get started with our free free resume templates. Paper? Download the one that best suits your experience, and get started writing. Finally, you can use our free industry resume samples for inspiration from similar job roles. More people need to phd thesis read this; too many times I’m still seeing job description oriented resumes and Objective paragraphs that are written in defense paper, the manner of essex phd thesis those written back in the sixties.

Glad you agree! Applicants really need to focus on selling themselves in write essay, the professional experience section rather than just including a laundry list of mundane duties and responsibilities. Essex Phd Thesis? Thanks for your comment! When referring to your budget amount, do you put the revenue or do you put the write a descriptive essay, bottom line, i.e. net income? HR managers like to see that an applicant was able to accomplish their goals within a set a budget. The ability to phd thesis manage a budget also instills confidence in the employer that you are a trustworthy and responsible candidate. Good luck on the job hunt! I’m trying to reports high school revamp my resume in a format that focuses more on accomplishments than tasks, but there are always going to be key parts of past jobs that simply don’t lend themselves to phd thesis being quantified. What’s the best approach to cover letter making sure this information gets in without losing its importance? Also, some of my experiences were quantifiable, but only essex, tacitly.

For example, I helped co-workers by papers on ups automating some of their tasks, which saved them time when doing them. I have no idea how much time, but I know it was significant. Phd Thesis? What kind of terminology should I use to term papers on ups convey these kind of achievements? Great question! Many job seekers would agree that it’s not always easy to quantify their achievements. In your case, the next best thing would be to estimate the amount of time you saved your co-workers. As long as you are realistic with your estimate there is no harm in including it on essex phd thesis your resume.

If you say that you saved your co-workers an average of 5 hours a day, that might raise flags with hiring managers. However, an defense paper, estimate of 15-20 minutes a day is reasonable and still quite impressive. The most important thing to consider is that if HR calls your previous manager, he/she will back up your claim. Essex Phd Thesis? If so, then you are in the clear. Good luck on the job hunt!

So my dilemma is that I don’t have quantifiable accomplishments. Meaning – I can list what I did for the company, but the dissertation, company will not verify it. Essex? The management turnover rate is so ridiculously high that they don’t really know who implemented it or when it was implemented. And so many people take credit for what someone else did, that it is thesis defense, difficult to actual say, “Yes, so and so did do that.” How do you include that in a resume? Is it possible? Even if you don’t have quantifiable achievements, you can still find ways to add numbers to your resume. Think of ways to phd thesis quantify your job duties: Did you handle a budget? Train new employees?

How many clients did you handle per month? How large was the team you worked with? These are just a few ways you can include numbers in editing english, your resume without having major accomplishments. Essex Phd Thesis? Try to get creative and cover brainstorm other ways to essex quantify your resume. Good luck on the job hunt, Since you have a lot of michael dissertation educational experience, we suggest checking out our education section writing guide: You’ll have to highlight all your accomplishments that you have achieved in your previous jobs.

Even if they seem small, including them on phd thesis your resume will show that you are achievement oriented. Also, if you ever helped train new employees, make sure you include that as well. It will demonstrate your ability to teach. Best of editing english luck! Wow! Congratulations for essex phd thesis, the great job you have done! You didn`t leave anything out; you even included a list of action verbs to describe work experience, and classified by skills. There`s a lot of reading to write essay of packingtown do, but sure it is phd thesis, worthwhile.

Glad you found it useful! Good luck on thesis defense the job hunt. This is unbelievable! Thanks for sharing this wealth of information. Phd Thesis? I am forever grateful! I just relocated to South America and would love to teach at a private elementary or secondary school, but didn’t know how to sell myself without a teaching certificate (I have a BA in editing english, Cultural Anthropology/Sociology), now I do, thanks to your team! Glad you found it helpful.

Good luck in phd thesis, South America! Some great advice on this page. On Ups? I already thought my resume was good but these tips will help take it to the next level. That’s a good point, but there are infinite reasons as to why someone might be looking for a new job even if they have achieved success in their current position. (bad management, relocating, not developing any new skills, etc) Thanks for the comment! Thank you!

Super helpful #128512; this may sound silly, but i didn’t realize how much i was doing for the company i work for essex phd thesis, until i read this. thank u. Not silly at all, we actually get that a lot. Most people are surprised to block cover letter learn how much they can add to their resume. Best of luck! Share How to Write an Achievement Oriented Resume Our code geeks and HR experts are proud to introduce our new Free Resume Builder software to help you land more interviews in today’s competitive job market.

We provide HR-approved resume templates, built-in job description bullet point phrases to choose from, and easy export to phd thesis MS Word and PDF. Get awesome job opportunities sent directly to your inbox. By clicking Send Me Job Alerts, I agree to the Resume Genius Terms of write a descriptive of packingtown Use and phd thesis Privacy Policy. Play the One-Minute Game That’ll Show You How to Improve Your Resume. Think you can judge the quality of a resume within 6 seconds? The answer may surprise you. Put your skills to the test, and editing english learn how to make your resume 6 second worthy!

3 Reasons Why I Wouldn't Hire Tom Brady. Tom Brady’s resume is a couple yards short of essex phd thesis a touchdown. Term? There are tons of errors throughout. See why. How to Modify and Maximize your Resume Template. Need a resume template? Feel free to download one, but be sure to make small modifications to unlock your. Would You Rather Work for a Man or a Woman? Do people still care whether they work for a man or woman, or do most people simply look for a nice job. Resume Builder Comparison | Resume Genius vs. LinkedIn Labs.

What are the differences between the major online resume builders? Here's an in depth analysis of what. Resume Genius' builder, resources, advice and career tips are regularly featured on some of the essex phd thesis, world's leading online and jerry michael hawkins dissertation offline publications including: Our code geeks and HR experts are proud to introduce our new Free Resume Builder software to help you land more interviews in today's competitive job market. HR-proven resume templates, built-in job description bullet point phrases to phd thesis choose from, and easily export to MS Word and PDF.

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an phd thesis, element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group.

Class field theory describes the abelian extensions of term papers on ups, a number field in terms of the arithmetic of the field. Essex. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008). Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at Please send comments and corrections to me at the address on my web page. The photograph is of the creative book reports high school, Fork Hut, Huxley Valley, New Zealand.

Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essex. . . . . Write Essay Of Packingtown. . . . . . Essex. 5 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editing English. . . . . . . . . . 5 Acknowledgements . . . Phd Thesis. . . Block Letter. . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . . . Jerry Michael Hawkins Dissertation. . . . Phd Thesis. . . . 5 Introduction . . . A Descriptive. . . . . . . . . . . . . . . . . Essex Phd Thesis. . . . . . . . Jerry Michael Dissertation. . . . . . . . . . . . 1 Exercises . . . . . . . . . . . . . . . . Essex Phd Thesis. . . . Book High. . . . . . . Essex Phd Thesis. . . . . . . . . . . . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . Defense Paper. . . . . . . . . . . . . . . . . . . . Essex. . . . . . . . . Hawkins Dissertation. . . 7 Ideals in products of rings . . Essex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Noetherian rings . . . . . . . . . . . . Jerry. . . . . . Phd Thesis. . . . . . . . . . . Michael Dissertation. . . . . . . . . 8 Noetherian modules . . . Essex Phd Thesis. . . . . . . . . . . . Cover Letter. . . . . . Essex Phd Thesis. . . . . . . . . . . . . . . 10 Local rings . . . . On Ups. . . . . . . . . Essex. . . . Block Cover. . Essex Phd Thesis. . . . Book School. . . . . . . . . . . . . . . . . . Phd Thesis. . 10 Rings of fractions . . . . . . . . . . . . . . . Jerry Michael. . . . . . . . . . . . . . Phd Thesis. . . . . . . 11 The Chinese remainder theorem . . . Term. . . . . . . . . Essex. . . . . . . . . . . . . . . . 12 Review of tensor products . . . . . . . . . Reports High School. . . . . . . . . . . . . . . Phd Thesis. . . . . . Michael Hawkins Dissertation. . . 14 Exercise . . . . . . . . . . . Phd Thesis. . . . . . Dissertation. . Essex Phd Thesis. . . Jerry Michael Dissertation. . . Essex. . . . . . . . . . . . Paper. . . . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . . Essex Phd Thesis. . . . . . . . . . . . . . . 19 Dedekind’s proof that the integral elements form a ring . . . . . . . Editing English. . . . . . . Essex Phd Thesis. . 20 Integral elements . . . . . . . . . . . . . . . . . . . . . . . . Creative Book Reports High School. . . . . . . . . . . 22 Review of bases of A-modules . . . . . . . . . . . . . . . . . . . . . . . Essex Phd Thesis. . . . . 25 Review of norms and traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissertation. . 25 Review of bilinear forms . . . . . . . . . . Essex. . . . . . . . . . . . Term On Ups. . . . . . . . . . 26 Discriminants . . . . . . . . . . . . . . . . . . . . Essex. . . . . . . . . Write A Descriptive. . . . . . . . . 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . Essex. . . . . . Editing English. . 29 Finding the ring of integers . . Phd Thesis. . . . . . . . . . Term On Ups. . . . . . . . . . . . . . . . . . . 31 Algorithms for finding the ring of integers . . . . . . . . . . . . . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essex. . 38.

3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . Jerry Michael Hawkins. . . . . . . . . . . 40 Dedekind domains . . . . . . . . . . . . Essex. . . . . . . . Cover Letter. . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . . Essex. . . . . . . Term Papers On Ups. . . . . . . . . . . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essex. . . 46 Discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block Cover. . 49 Integral closures of Dedekind domains . . . Essex. . . . . . . . . Creative Reports. . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . Essex Phd Thesis. . . . . . . . Editing English. . . . . Phd Thesis. . . . . . 52. Factorization in extensions . . . . . . . . . . . . . Write A Descriptive Essay. . . . . . . . Phd Thesis. . . . . . . . . Defense. . 52 The primes that ramify . Essex. . . . . . . . . Book High. . . . . . . . . . . Essex. . . Write A Descriptive Of Packingtown. . Phd Thesis. . . . . Term Papers On Ups. . . . . . 54 Finding factorizations . . . . . . . . . Phd Thesis. . . . . . . . . . . . Michael Hawkins. . . . . . . . . . . . . 56 Examples of factorizations . . . . . . . . . . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . 57 Eisenstein extensions . . . . . . Write Essay Of Packingtown. . . . . . . . Phd Thesis. . . . . . . . . . . . . . . . . . . . Editing English. 60 Exercises . . . . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Letter. . . 63 Statement of the essex, main theorem and its consequences . Michael Hawkins Dissertation. . . . . . Phd Thesis. . . . . . . . . . . 65 Lattices . . . . Book High. . . . . . . Essex. . . . . Jerry. . . . . . . . . . . . . . Essex. . . . . . . . Write A Descriptive. . . . . . 68 Some calculus . . . . . . . . . . . . Essex Phd Thesis. . . . . . . . . . . . Jerry. . . . . . . . . Essex. . . . . . 73 Finiteness of the class number . . . . . . . . . . . . . . . . . . . . . . . . . Write. . . 75 Binary quadratic forms . . . Essex Phd Thesis. . . . . . . . . . . . . Thesis Paper. . . . Phd Thesis. . . . . . Jerry Dissertation. . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phd Thesis. . . . . Editing English. . . . . Phd Thesis. 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . Jerry Michael. . . . . . . . . Essex. . . . . . . . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . . Cover. . Phd Thesis. . . . . . Editing English. . . . . . . . . . . Essex Phd Thesis. 82 Computation of the rank . . . . . . Editing English. . . . . . . Essex. . . . . . . . . . . . Editing English. . Phd Thesis. . . . . Block. . . 83 S -units . . . . Phd Thesis. . . . . . . . . . . . . . . . . . . Thesis Defense Paper. . . . . . . . . . . . . . . . . . . Essex Phd Thesis. 85 Example: CM fields . . . . . . . . . . . . Write A Descriptive. . . . . . . . . . . . . . . . . . . . Essex Phd Thesis. . . 86 Example: real quadratic fields . Term. . . . . . . . . . . Essex. . . . . . . . . . . . . . . . . 86 Example: cubic fields with negative discriminant . . . . . . . . . Jerry Dissertation. . Essex Phd Thesis. . . . . . . . 87 Finding .K/ . . . Michael Hawkins Dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . . . . 89 Regulators . . . . . Phd Thesis. . . . . . . . . . . . . . . . . . . . . . . Michael Dissertation. . . Essex Phd Thesis. . . . . . . . . . 89 Exercises . . . . . Book Reports School. . . . . . . . . . . . . . . . . . . Phd Thesis. . . . Papers On Ups. . . . . . . . . . . . . 90. 6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . Essex. . . . . . . . . . . . . . . . . . . Paper. . . . . . . . . . 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . Phd Thesis. . . School. . . . . . . . . . . . . . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . . . . . . . . Essex. . 98 Exercises . . . A Descriptive Essay. . . . . . . . . . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . . . . Reports High School. . . Essex. . . 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . . . . . . . Cover Letter. . . Essex Phd Thesis. . . . . . . . . Editing English. . . . Essex Phd Thesis. . . . . . . . Thesis. . . . Phd Thesis. . . 101 Nonarchimedean valuations . . . . . . . . . Papers On Ups. . . Essex. . . . . Book Reports. . . Essex. . . . . . . Term Papers. . . . . Essex Phd Thesis. . Editing English. . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . . . . . Phd Thesis. . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . Write Essay Of Packingtown. . . . . . . . . . . 105 Complete list of valuations for the rational numbers . . . . . . . . . . . . . . . . 105 The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . Phd Thesis. . . . . . . . . . . . . . . . . . . . . . 109 Completions . . . . . . . . A Descriptive Essay. . . . . . . . Phd Thesis. . . . . Book School. . . . . . . . . . . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . Essex. . . . . . Jerry Hawkins Dissertation. . . . . 111 Newton’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.

Newton’s polygon . . Essex Phd Thesis. . . . . . . . . . . . . . . . . . Hawkins Dissertation. . . . . . . . . . Phd Thesis. . . . . . . 120 Locally compact fields . . . . . . . . . . . . . . . Term. . . . . . . . . . . . . . . . . 122 Unramified extensions of a local field . . . . . . . . . . . Phd Thesis. . . . . . . Hawkins Dissertation. . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Ramification groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phd Thesis. . . . . 126 Krasner’s lemma and applications . . . . . . . . . . Editing English. . Essex Phd Thesis. . . . . . . . . . . . . . . 127 Exercises . . . Letter. . . Phd Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editing English. . . . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . . . . . . . . . . . Phd Thesis. . . . . . . 131 The product formula . Term Papers On Ups. . . . . . . . . . . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . . . 133 Decomposition groups . . . . Thesis Defense Paper. . Essex Phd Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . Term Papers. . Phd Thesis. . Papers. . . . . . . . . . . . . . . . . Phd Thesis. . . . . . 137 Examples . . . . . . . . . . . . Reports High. . . . . . . . . Essex. . . . . . . . . Editing English. . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . Essex Phd Thesis. . . . . . . . . . . . . . . Editing English. . . . 140 Computing Galois groups (the easy way) . . . Essex Phd Thesis. . . . On Ups. . . . . . . . . . . . . . . . . 141 Applications of the Chebotarev density theorem . . . . . . . . . . . . . . . . . . 146 Exercises . . . . . . . . Phd Thesis. . . . . Reports High. . . . . . . . . . . . . . . . . . . . . . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of essex, integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number.

For integers m and n, mjn means that m divides n, i.e., n 2mZ. Jerry Hawkins Dissertation. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in phd thesis, S when S is finite). Let I and A be sets; a family of elements of A indexed by editing english I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. Phd Thesis. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q. The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at FT Fields and Galois Theory, v4.22, 2011.

GT Group Theory, v3.11, 2011. Jerry Michael Hawkins Dissertation. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for phd thesis earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?. The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works.

EULER (1707–1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the term papers on ups, quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to phd thesis a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the write a descriptive essay, class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of phd thesis, rank rC s?1. KUMMER (1810–1893). Papers On Ups. He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for regular primes. HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions.

EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum). RIEMANN (1826–1866). Studied the essex, Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the editing english, correct definition of the ring of integers in essex phd thesis, a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into block letter, finitely many classes.

Defined the zeta function of a number field. WEBER (1842–1913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941). He gave the first definition of the field of p-adic numbers (as the essex, set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943).

He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. High School. TAKAGI (1875–1960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields.

HECKE (1887–1947). Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions. ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979). He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (1906–1998).

Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the phd thesis, earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925– ). He proved new results in thesis defense, group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is phd thesis said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields.

Three problems present themselves. First, factorization in a field only makes sense with respect to on ups a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by phd thesis how much it fails. Finally, since factorization is book reports school only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is essex phd thesis a square-free integer. The. Essay Of Packingtown. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. Essex Phd Thesis. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and letter, so 1=2C. is an algebraic integer in Q? p 5?.

Let d be a primitive d th root of essex phd thesis, 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P. as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is a descriptive essay not a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is essex phd thesis a unique factorization domain if every nonzero element of write a descriptive essay, A can be expressed as a product of irreducible elements in phd thesis, essentially one way. Is the ring of integers OK a unique factorization domain? No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is on ups true for phd thesis all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have.

6D 2 3D .1C p ?5/.1?. Creative Reports. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and phd thesis, no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and. 1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in block, OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is unique if all irreducible elements are prime. What can we recover?

Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of phd thesis, Kummer and Dedekind was to michael dissertation enlarge the set of “prime numbers” so that, for example, in Z? p ?5? there is essex phd thesis a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”. How do we define “ideal factors”? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”.

Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for term on ups all b 2OK I. Essex. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Defense. Since every gen- erator is divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and phd thesis, so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. Creative Reports High. I further claim that the essex phd thesis, four ideals. Defense Paper. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime. For example, the phd thesis, obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so.

Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index. A Descriptive Essay Of Packingtown. The index is called the class number hK of K. We shall see that. hK D 1 ” OK is a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units.

For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Essex. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and creative school, it was recognized that a stumbling block to phd thesis proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to term papers prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that. Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction.

Liouville immediately questioned a step in Lame?’s proof in essex, which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on write essay the arithmetic of the fields Q?? to prove Fermat’s last theorem for all regular primes, i.e., for all primes p such that p does not divide the class number of phd thesis, Q?p?. Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and thesis defense paper, Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for essex fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY. Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. Creative Book High School. The notes in Narkiewicz 1990 document the origins of most significant results in essex, algebraic number theory.

Lemmermeyer 2009, which explains the origins of “ideal numbers”, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from book high, Commutative. Many results that were first proved for rings of integers in number fields are true for essex more general commutative rings, and it is more natural to prove them in that context.1. All rings will be commutative, and have an write essay, identity element (i.e., an element 1 such that 1a D a for all a 2 A), and phd thesis, a homomorphism of rings will map the identity element to the identity element. A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X.

ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of on ups, A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and phd thesis, it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an editing english, ideal in A — in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA! A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of essex phd thesis, A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and editing english, only if the quotient ring A=a is an integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb.

An ideal m in phd thesis, A is maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an editing english, ideal a ¤ A containing m but distinct from essex phd thesis, it. An ideal a is maximal if and only if A=a is a field. Letter. Every proper ideal a of A is contained in a maximal ideal — if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorn’s lemma. In particular, every nonunit in phd thesis, A is contained in a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in products of term, rings.

PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in AB , and every ideal in AB is of this form. The prime ideals of AB are the ideals of the form. pB (p a prime ideal of A), Ap (p a prime ideal of essex, B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Creative Reports. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and phd thesis, hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and reports high school, only if there is a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j: A ring A is Noetherian if every ideal in A is finitely generated.

PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. (c) (a): Let a be an ideal in essex, A, and let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/. If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant.

PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements. Jerry. PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ ” bja, with equality if and phd thesis, only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Editing English. Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and phd thesis, the ideals can’t be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the write, proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated. PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in phd thesis, M has a maximal element. PROOF.

Similar to the proof of Proposition 1.3. 2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of michael dissertation, M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and essex, if N is Noetherian, the intersection of the chain with N becomes constant. Now the claim shows that the chain itself becomes constant. Michael Dissertation. 2. PROPOSITION 1.8 Let A be a Noetherian ring. Then every finitely generated A-module is Noetherian.

PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of phd thesis, generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in editing english, the lemma is essential. Phd Thesis. For example, if A is a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions. Let A be an defense paper, integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. Essex. For example, Q is the field of fractions of Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define.

S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg: 2This notation conflicts with a later notation in which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of S?1A aec D a if a is write a descriptive a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in S?1A. For the phd thesis, reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of a=s or s is in p (because it is prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the michael dissertation, set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF.

It is easy to see that. Essex. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse. 2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). Jerry. (b) We list the essex phd thesis, prime ideals in some rings: Note that in general, for t a nonzero element of an integral domain, fprime ideals of creative book reports school, Atg $ fprime ideals of A not containing tg. fprime ideals of A=.t/g $ fprime ideals of essex, A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for editing english any integers x1; . ;xn, the congruences. The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is phd thesis one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and write, n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals.

Q .mi /. If the mi are relatively prime in. pairs, then mD Q mi , and so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence. PROOF. Suppose first that n D 2. Phd Thesis. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. Block Cover. ai Cbi D 1, all i 2: The product Q Cbi /D 1, and lies in a1C. Phd Thesis. Q i2 ai , and so.

We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2. We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to editing english A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in essex phd thesis, pairs, and let M be an A-module.

There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ! P A-linear. The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute).

We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products. It follows that if M and term papers on ups, N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is a vector space over k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that.

REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. Recall that M 0?N 0 is generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it. 2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. Essex Phd Thesis. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module.

Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and defense, M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism.

M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS. If A! B is an phd thesis, A-algebra and M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is book school a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS. If f WA! B and gWA!

C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by phd thesis the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X? .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in cover letter, (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. It was not until the essex phd thesis, 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and michael hawkins, Emmy Noether). Embed this document on your website.

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Essay on Marriage: Meaning, Functions and Forms. Here is your essay on marriage, it’s meaning, functions and forms! Marriage and family sociologically signifies the stage of essex greater social advancement. It is book reports high school, indicative of man’s entry into the world of emotion and feeling, harmony and culture. Long before the institution of marriage developed, man and woman may have lived together, procreated children and died unwept and unsung. Their sexual relations must have been like birds and animals of momentary duration.

Image Courtesy :–Zyo4IJENYk/TtuIdHcjm-I/AAAAAAAACwQ/Rg89UBiMU0A/s1600/lklk.jpg. Marriage as an institution developed over essex the time. It may have been accepted as a measure of social discipline and as an expedient to eliminate social stress due to the sex rivalry. The growing sense and sensibility may have necessitated the acceptance of norms for formalising the union between man and editing english, woman. Marriage is the most important institution of human society. It is a universal phenomenon. It has been the backbone of human civilisation. Human beings have certain urges like hungers, thirst and sex. Society works out certain rules and regulation for satisfaction of these urges. The rules and regulations, which deal with regulation of sex life of human beings, are dealt in the marriage institution.

We can say that the Marriage is as old as the institution of family. Both these institutions are vital for the society. Family depends upon the Marriage. Marriage regulates sex life of human beings. Marriage creates new social relationships and reciprocal rights between the spouses. It establishes the rights and the status of the essex phd thesis, children when they are born. Each society recognises certain procedures for creating such relationship and rights. The society prescribes rules for prohibitions, preferences and prescriptions in deciding marriage. It is this institution through which a man sustains the continuity of his race and attains satisfaction in a socially recognised manner. Sociologists and anthropologists have given definitions of marriage. Some of the important definitions are given below.

Edward Westermark. “Marriage is a relation of one or more men to one or more women which is recognised by custom or law and involves certain rights and duties both in the case of the parties entering the union and in the case of the children born of it. As B. Malinowski defines, “Marriage is write essay of packingtown, a contract for the production and maintenance of children”. According H.M. Johnson, “Marriage is a stable relationship in which a man and a woman are socially permitted without loss of standing in essex phd thesis, community, to have children”. Ira L. Reiss writes, “Marriage is a socially accepted union of individuals in editing english, husband and wife roles, with the key function of legitimating of parenthood”. William Stephens, the anthropologist, says that marriage is: (1) A socially legitimate sexual union begun with. (2) A public announcement, undertaken with. (3) Some idea of performance and assumed with a more or less explicit. (4) Marriage contract, which spells out reciprocal obligations between spouses and between spouses and their children. William J. Goode, the famous family sociologist has tried to combine the two objectives of marriage i.e. to regulate sex life and to essex recognize the newborn.

It was perhaps for this reason that American sociologists came out with the statement that no child should be born without a father. Although different thinkers have tried to provide definition of term papers marriage, but there is no universally acceptable definition of marriage. There seems to be, however, a consensus that marriage involves several criteria that are found to essex exist cross-culturally and throughout time. For example, Hindu marriage has three main objectives such as Dharma, Progeny and Sexual Pleasure. Individual happiness has been given the least importance. It is considered to be sacrament, a spiritual union between a man and a woman in the social status of husband and wife. In Western countries, marriage is a contract.

Personal happiness is given the utmost importance. People enter into matrimonial alliances for the sake of seeking personal happiness. If this happiness is-not forthcoming they will terminate the relationship. Marriage is thus cultural specific. The rules and regulations differ from one culture to another. We can, however, identify certain basic features of this institution. (1) A heterosexual union, including at write a descriptive essay least one male and phd thesis, one female. (2) The legitimizing or granting of approval to the sexual relationship and the bearing of children without any loss of standing in the community or society. (3) A public affair rather than a private, personal matter. (4) A highly institutionalized and patterned mating arrangement. (5) Rules which determine who can marry whom.

(6) New statuses to man and woman in the shape of husband and wife and father and mother. (7) Development of personal intimate and affectionate relationships between the spouses and parent and children. (8) A binding relationship that assumes some performance. The above discussion helps us to conclude that the boundaries of thesis paper marriage are not always precise and clearly defined. Phd Thesis? It is, however, very important institution for the society as it helps in paper, replacement of old and dying population. Marriage is an institutionalized relationship within the family system. It fulfills many functions attributed to essex phd thesis the family in general. Family functions include basic personality formation, status ascriptions, socialization, tension management, and replacement of members, economic cooperation, reproduction, stabilization of adults, and the like. Many of these functions, while not requiring marriage for their fulfillment, are enhanced by the marital system”. In fact, evidence suggests marriage to write be of great significance for the well-being of the individual. Essex? Researchers have shown that compared to the unmarried, married persons are generally happier, healthier, less depressed and disturbed and less prone to premature deaths.

Marriage, rather than becoming less important or unimportant, may be increasingly indispensable. The functions of marriage differ as the editing english, structure of marriage differs. ‘For example, where marriage is phd thesis, specially an extension of the kin and extended family system, then procreation, passing on the family name and continuation of property become a basic function. Thus, to not have a child or more specifically, to not have a male child, is sufficient reason to replace the editing english, present wife or add a new wife. Where marriage is based on “free choice,” i.e. parents and kinsmen play no role in selecting the partner, individualistic forces are accorded greater significance. Thus in the United States, marriage has many functions and involves many positive as well as negative personal factors : establishment of a family of one’s own, children, companionship, happiness, love, economic security, elimination of loneliness etc. The greater the extent to which the perceived needs of phd thesis marriage are met, and editing english, the fewer the essex, alternatives in the replacement of the unmet needs, the greater the likelihood of marriage and the continuation of that marriage. At a personal level, any perceived reason may explain marriage, but at a social level, all societies sanction certain reasons and renounce others.

Societies evolved mannerism and method for selection of the spouses, according to their peculiar socio-economic and political conditions, and in defense, accordance with their levels of cultural advancement. This explains on the one hand the origin of the various forms, of marriage and on the other the differences in the attitude of societies towards the institution of marriage. Some have accepted it as purely a contractual arrangement between weds, while others hold it as the sacred union between man, and woman. Forms of marriage vary from phd thesis society to society. Marriage can be broadly divided into two types, (1) monogamy and (2) polygamy. Monogamy is jerry hawkins dissertation, that form of marriage in which at a given period of time one man has marital relations with one woman.

On the death of the spouse or one of the partners seek divorce then they can establish such relationship with other persons but at a given period of time, one cannot have two or more wives or two or more husbands. This one to one relationship is the most modern civilized way of living. In most of the societies it is this form, which is essex phd thesis, found and recognized. Write A Descriptive Essay Of Packingtown? It should be noted that on a societal basis, only about 20 per cent of the societies are designated as strictly monogamous, that is, monogamy is the required form. When monogamy does not achieve stability, certain married persons end their relationship and remarry. Thus, the second spouse, although not existing simultaneously with the essex, first, is sometimes referred to as fitting into a pattern of sequential monogamy, serial monogamy or remarriage. Keeping in view the advantages of monogamy the thesis defense, world has granted recognition to monogamous form of marriage. The following are its advantages:

1. Better Adjustment: In this form of marriage men and women have to adjust with one partner only. In this way there is better adjustment between them. 2. Greater Intimacy: If the number of essex people in the family will be limited there will be more love and affection in the family. Because of which they will have friendly and deep relations. 3. Better Socialization of Children: In the monogamy the children are looked after with earnest attention of block parents. The development of modes of children will be done nicely. There will be no jealously between the parents for essex, looking after their children.

Family happiness is maintained under monogamy which is completely destroyed in other forms of essay marriage because of jealousy and essex phd thesis, other reasons. Thus, in this form of marriage, family is defined as happy family. 5. Equal Status to Woman: In this form of marriage the status of woman in family is equal. If husband works she looks after the house or both of them work for strengthening the economic condition of the family. 6. Equalitarian way of block cover Living: It is only under monogamous way of living that husband and wife can have equalitarian way of essex life. Under this system husband and wife not only share the familial role and obligations but also have joint decisions. The decision making process becomes a joint venture. 7. Population Control:

Some sociologists have the view that monogamy controls the population. Because of one wife children in the family will be limited. 8. Better Standard of Living: It also affects the standard of living within limited resources. One can manage easily to cover letter live a better life. It helps in the development of independent personality without much constraint and pressure. 9. Respect to old Parents: Old parents receive favouring care by essex, their children but under polygamy their days are full of bitterness. 10.

Law is in favour: Monogamy is legally sanctioned form of marriage while some are legally prohibited. 11. More Cooperation: In such a family there is close union between the couple and the chances of conflict are reduced and there is cooperation between husband and wife. It is editing english, more stable form of marriage.

There is better division of property after the essex phd thesis, death of parents. Monogamy is a marriage between one husband and one wife. So if the partner is not of choice then life loses its charm. On Ups? They have to adjust between themselves but now-a-days divorce is the essex phd thesis, answer to their problem. According to Sumner and editing english, Keller, “Monogamy is monopoly.” Wherever there is monopoly, there is phd thesis, bound to be both ‘ins and outs’. Some inpatients can’t have kids or some barren cannot have kids. If one of the partners has some problem couples cannot have children.

They have to suffer from childlessness. 4. Economic Factors: Marriage in creative reports high school, monogamy does not play part of income. Essex Phd Thesis? They have to write a descriptive depend upon their own occupation for living. If they are poor they will remain poor. So monogamy effects the economic condition of man and woman. 5. Better status to essex phd thesis Women: Monogamy provides better status to editing english women in essex phd thesis, the society. Editing English? They are counted equal to men. Some people do not like this form of marriage. When they do not get partner of their own choice they start sexual relations with other people.

This also leads to the problem of prostitution. Distinguished from monogamy is polygamy. Phd Thesis? Polygamy refer to the marriage of several or many. Polygamy is the form of marriage in which one man marries two or more women or one woman marries two or more men or a number of men many a number of women. According to editing english F.N. Balasara, “The forms of marriage in which there is plurality of partners is called polygamy”.

Polygamy, like other forms of essex phd thesis marriage is highly regulated and write, normatively controlled. It is likely to be supported by the attitudes and values of both the sexes. Essex Phd Thesis? Polygamy itself has many forms and jerry hawkins, variations. Polygamy is of three types: (i) Polygyny, (ii) Polyandry and (iii) Group marriage. Let us now discuss forms of polygamy in details, Polygyny is a form of marriage in which a man has more than one .wife at a time.

In other words it is a form of essex phd thesis marriage in which one man marries more than one woman at a given time. It is the prevalent form of block cover letter marriage among the tribes, Polygyny also appears to be the privilege of the wealthy, in many African societies the rich usually have more than one wife. This type of marriage is phd thesis, found in Ghana, Nigeria, Kenya and Uganda. In India, polygyny persisted from the Vedic times until Hindu Marriage Act, 1955. Now polygyny is visible among many tribes of India. Viewing polygyny cross-culturally, poiygynous families evidence specific organisational features: 1. In certain matters, sex particularly, co-wives have clearly defined equal rights. 2. Each wife is set up in a separate establishment. 3. The senior wife is given special powers and privileges. It has been suggested that if co-wives are sisters, they usually live in the same house; if co-wives are not sister, they usually live in separate houses. It is believed that sibling can better tolerate, suppress and thesis, live with a situation of sexual rivalry than can non-siblings.

Polygyny may be of two types: (i) Sororal polygyny and (ii) Non-soraral polygyny. Sororal polygyny is one in essex phd thesis, which all the wives are sisters. Non-sororal polygyny means the marriage of one man with many women who are not sisters. 1. Disproportion of sexes in the Population: When in any tribe or society male members are less in number and females are more, then this type of marriage takes place. 2. Out-migration of block cover letter male Population: To earn the livelihood male members migrate from essex phd thesis one society to another. This way there is a decrease in the number of males than females and polygyny takes place. Hypergamy also gives rise to polygyny.

Under this system the editing english, parents of lower castes or classes want to improve their social status by marrying their daughters in the higher caste or classes. 4. Desire for male Child: Among the primitive people importance was given to make children than females. Phd Thesis? Thus man was free to have as many marriages as he liked on the ground to editing english get male children. In some societies number of wives represented greater authority and status.

Particularly the leaders of primitive society increased number of wives in order to prove their superiority. A single marriage was considered a sign of poverty. So where marriage is taken as sign of essex phd thesis prestige and prosperity the editing english, custom of polygyny is natural. 6. Phd Thesis? Economic Reason: Where the people of the poor families were unable to find suitable husbands for their daughters they started marrying their daughters to rich married males. 7. Variety of Sex Relation: The desire for variety of sex relations is another cause of polygyny. The sexual instincts become dull by more familiarity.

It is stimulated by write essay, novelty. 8. Enforced Celibacy: In uncivilized tribes men did not approach the women during the period of pregnancy and while she was feeding the child. Thus long period of enforced celibacy gave birth to essex phd thesis second marriage. In uncivilized society more children were needed for agriculture, war and status recognition. Moreover, in some tribes the birth rate was low and death rate was high. In such tribes polygyny was followed to obtain more children. 10.

Absence of children: According to Manu, if wife is unable to high school have children, man is permitted to have more marriages. He further says if a wife takes her husband then he should live with her one year and take another wife. 11. Religious Reasons: Polygyny was permitted in phd thesis, the past if wife was incapable of forming religious duties in her periodic sickness because religion was given significant place in social life. 12. Patriarchal Society: Polygyny is found only in the patriarchal society where more importance is given to males and male member is the head of the family. (1) Better status of block cover letter children: In polygyny children enjoy better status.

They are looked after well because there are many women in the family to care. (2) Rapid growth of Population: In those societies where population is very less and birth rate is almost zero, for those societies polygyny is best suited, as it increases the population at faster rate. (3) Importance of Males: In polygyny males occupy higher status. More importance is given to husband by several wives.

(4) Division of phd thesis Work: In polygyny there are several wives. Therefore, there is a proper division of work at home. (5) Variety of Sex Relations: Instead of going for extra marital relations husband stays at home because his desire for variety of sex relations is fulfilled within polygyny. (6) Continuity of Family: Polygyny came into existence mainly because of jerry michael hawkins inability of a wife to produce children. Polygyny provides continuity to the family tree. In absence of one wife other women in the family produce children. 1. Lower status of Women:

In this form of marriage women have very low status; they are regarded as an phd thesis, object of pleasure for their husbands. They generally do not have a right to take decisions about their welfare; they have to depend upon their husband for fulfillment of editing english their basic needs. 2. Jealousy as stated by Shakespeare: “Woman thy name is jealousy”. When several wives have to share one husband, there is bound to be jealousy among co-wives. Jealousy leads to inefficiency in their work. They are not able to socialize their children in a proper manner in such atmosphere. 3. Low Economic Status: Polygyny increases economic burden on the family because in many cases only husband is the bread winner and whole of the phd thesis, family is dependent on him. 4. Population Growth: This type of marriage is harmful for developing society and poor nations because they have limited resources Further increase in population deteriorates progress and development of that society.

5. Fragmentation of Property: In polygyny all the children born from different wives have share in father’s property. Jealousy among mothers leads to property conflicts among children as a result property is creative reports, divided and income per capita decreases. 6. Essex? Uncongenial Atmosphere: Polygyny does not promise congenial atmosphere for the proper growth and development of children. There is lack of affection among the members. As such families have large number of members. They fail to provide proper attention to all of editing english them.

This gives rise to many immoral practices in essex, the society. It is a form of reports high school marriage in which one woman has more than one husband at a given time. According to K.M. Kapadia, Polyandry is a form of union in which a woman has more than one husband at a time or in phd thesis, which brothers share a wife or wives in common. This type marriage is prevalent in few places such as tribes of Malaya and some tribes of India like Toda, Khasi and Kota etc.

Polyandry is paper, of two types: (i) Fraternal Polyandry and. (ii) Non-Fratemai Polyandry. (i) Fraternal Polyandry: In this form of essex polyandry one wife is regarded as the wife of all brothers. All the reports school, brothers in a family share the same woman as their wife. The children are treated as the offspring of the eldest brother, it is found in some Indian tribes like Toda and Khasis. This type of marriage was popular in Ceylon (Srilanka at present). (ii) Non-Fraternal Polyandry: In this type of polyandry one woman has more than one husband who is not brothers.

They belong to different families. The wife cohabits with husbands in phd thesis, turn. In case of Fraternal Polyandry, the wife lives in the family of her husbands, while in case of non-fraternal polyandry, the wife continues to stay in the family of her mother. This type of polyandry is found among Nayars of Kerala. 1. Lesser number of Women:

According to Westermark, when the number of women is lesser than the number of write a descriptive essay males in a society, polyandry is found. For example, among Todas of Nilgiri. But according to Brifficult, polyandry can exist even when the phd thesis, number of women is not lesser e.g. in Tibet, Sikkim and michael hawkins dissertation, Laddakh polyandry is found even though there is not much disparity in the number of men and women. In some tribal societies female infanticide is phd thesis, present; as a result these female population is less than male population. Further males do not enjoy good status. Therefore, one female is creative book high school, married to a group of brothers and polyandry exists. 3. Matrilineal System: Just in contrast to phd thesis above noted point, it has also been argued that polyandry exists in matrilineal system where one woman can have relationship with more than one man and write, the children instead of phd thesis getting the name of father are known by papers on ups, mother’s name.

Polyandry exists in such areas where there is scarcity of natural resources. It is for essex phd thesis, this reason many men support one woman and her children. In societies where there is bridge price, polyandry exists. Brothers pay for one bride who becomes wife of paper all of them. 6. Division of Property: To check the division of ancestral property polyandry is favoured. When all the essex phd thesis, brothers have one wife then the question of division of property does not arise. 7. Production and on ups, labour: Polyandry not only essex phd thesis avoids division of property but it also increases production in agriculture.

All the brothers work together because they have to support only one family. Term Papers? Thus production and phd thesis, income increases, further there is no expenditure with regard to labour because all the husbands contribute their share of work. Polyandry exists in some societies mainly because of customs and term papers on ups, traditions of that particular society. Generally, polyandry is found in such areas which are situated far away from modern developed areas. (1) Checks Population Growth: It checks population growth because all the male members of the family share one wife. As a result population does not increase at that rapid rate, the essex, way in which it occurs in polygyny Therefore, it limits the size of the book reports, family. (2) Economic Standard: Polyandry helps to phd thesis unhold the economic standard of the family. It strengthens the economic position of the family because all the members work for papers, the improvement of the essex, family. (3) Greater Security:

With large number of males working after the family affairs, other members of the family especially women and children feel quite secure. Greater security among the members develop sense of we-feeling among the editing english, members of the family. (4) Property is kept Intact: In polyandry family does not get divided. The property of the family is held jointly and thus it is phd thesis, kept intact. (5) Status of Women: In polyandry one woman is editing english, wife of large number of husbands. As a result she gets attention of essex all the members and thus enjoys a good status in the family. She feels quite secure because in the absence of block cover letter one husband other males are there to fulfill her basic needs.

When all the men have to share one woman, family quarrels and tensions are ought to be there. Husbands feel jealous of one another which adversely effect congenial atmosphere of the family. When children have large number of fathers they fail to select appropriate model for essex phd thesis, themselves. This adversely effects their personality configuration. (3) Health of the Woman: It adversely effects health of a woman because she has to satisfy several husbands. It not only has negative effect on the physical health but also on mental health of the woman.

According to biologists if the same woman cohabits with several men, it may lead to sterility, further lack of sex gratification give rise to extra-marital relationship of husbands. In matrilineal system where polyandry is found husbands do not enjoy high status. They do not give their name to the children. (6) Lack of Attachment: In many tribes where polyandry exists husbands do not live permanently with their families. They are visiting husband who visit the family for a specific period. They do not get love and affection of their children because children feel unattached to their fathers. (7) Less Population: This form of marriage decreases population growth.

In some tribal societies where polyandry continues to exist may get extinct after a gap of few years. This is school, another outcome of phd thesis this practice. Group marriage is that type of marriage in which a group of men marry a group of women. Cover? Each man of male group is essex, considered to dissertation be the essex phd thesis, husband of every woman of female group. Similarly, every woman is the block cover, wife of every man of male group. Pair bonded or Multilateral marriage are the substitute term for group marriages. This form of marriage is found among some tribes of New Guinea and Africa. In India group marriage is essex phd thesis, practised by the Toda Tribe of Nilgiri Hills.

Except on jerry dissertation, an experimental basis it is an extremely rare occurrence and phd thesis, may never have existed as a viable form of marriage for any society in the world. The Oneida community of New York State has been frequently cited as an example of group marriage experiment. It involved economic and sexual sharing based on block letter, spiritual and religious principles. Like most group marriage on record, its time span was limited. Rarely do they endure beyond one or two generations.

Levirate and Sororate: In levirate the essex phd thesis, wife marries the brother of the dead husband. If a man dies, his wife marries the michael dissertation, brother of her dead husband. Marriage of the widow with the dead husband’s elder brother is called Senior Levirate. But when she marries to the younger brother of the dead husband, it is called Junior Levirate.

In Sororate the husband marries the sister of his wife. Sororate is again divided into two types namely restricted Sororate and simultaneous Sororate. In restricted sororate, after the death of one’s wife, the man marries the sister of his wife. In simultaneous sororate, the sister of one’s wife automatically becomes his wife. Concubinage is a state of living together as husband and wife without being married. It is essex phd thesis, .cohabitation with one or more women who are distinct from wife or wives. Concubinage is sometimes recognised by various societies as an accepted institution. A concubine has a lower social status than that of a wife. The children of a concubine enjoy a lower status in defense, the society. Leave a Reply Click here to cancel reply.

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