Powered by Blogger.

Why do we care about unique factorization?

Thursday, February 21, 2008

I know there's a keen expectancy out there for the next installment of our series on algebraic number theory – so here it is. Check here for preceding installments.

In the article before the most recent one, we reviewed the fact that there is unique factorization into primes in the ring ℤ of ordinary integers. And in the most recent article, we saw that some rings of algebraic integers do not have unique factorization.

But we still haven't explained why one might care about unique factorization. That's what we're going to take up now.

One may reasonably ask why this matters so much that we've spent so much time on it. The answer is that even leading mathematicians did not clearly appreciate, until the 1840s, how unique factorization could fail in rings of algebraic integers – and they sometimes based erroneous or incomplete proofs on the assumption of unique factorization.

The first noteworthy case of this seems to have been Leonhard Euler (1707-83). Number theory was one of a vast number of Euler's interests, and one of the problems he dealt with was what is now known as Fermat's Last Theorem: The equation xn+yn=zn has no nontrivial integer solutions for integers n>2. Pierre de Fermat (1601-65) himself publicly stated this result only in case n=3 or 4. Fermat stated the case for general n only in a private note in his copy of Diophantus' Arithmetica.

No written proof of this by Fermat in even the simplest cases is known, but it can easily be proven if n=4 by a technique Fermat invented, called the "method of infinite descent". This method involves assuming a solution exists for some triple (x,y,z) and then showing that there must always be another non-trivial solution (x′,y′,z′) in which at least one of the numbers is strictly smaller than any number of the original solution. Since that's impossible, the contradiction shows there couldn't have been any solution to begin with.

Anyhow, Euler knew the proof when n=4, and set out to give a proof along the same lines when n=3. He had the brilliant and original idea to work with numbers of the form a+b√-3 for a,b∈ℤ, that is, numbers in ℤ[√-3]. This was a great idea because it provided an entirely new and powerful set of tools for dealing with questions about ordinary integers. Unfortunately, as prescient as Euler was, there were too many subtleties in this area to use the tools correctly from the start.

One step of the argument involved reasoning that if for some c∈ℤ, c3 = a2+3b2 = (a+b√-3)(a-b√-3), and if the factors a+b√-3 and a-b√-3 are relatively prime, then each of the factors must themselves be cubes of numbers in ℤ[√-3]. One problem is that ℤ[√-3] isn't the full ring of integers of ℚ(√-3), since -3≡1 (mod 4). But even disregarding that, the conclusion depends on unique factorization of the numbers involved. Although it happens to be true that unique factorization holds in the integers of ℚ(√-3), Euler didn't seem to recognize the need to prove that.

This same lack of clarity about unique factorization in rings of algebraic integers seems to have persisted into the 1840s. In 1847 Gabriel Lamé (1795-1870) thought he had a proof of Fermat's theorem for arbitrary n. Lamé worked with numbers of the form ℤ[ζn] where ζn is a root of xn-1=0 – which is called an nth root of unity. (Here we assume n is the smallest integer for which the chosen ζn satisfies the equation.)

ℤ[ζn] is called the ring of cyclotomic integers (for a particular choice of n), and it is in fact the ring of integers of the field ℚ(ζn), the nth cyclotomic field – of which we shall have much to say later on. By 1847 some astute mathematicians did recognize the need for proof of unique factorization, and they pointed it out to Lamé. He must have quickly appreciated the problem, since he didn't persist in developing his "proof".

The purported proof went something like this: Suppose there were a solution of xn+yn=zn for some n>2 and integers x, y, and z that are relatively prime. The equation could be rewritten as
xn = zn - yn = ∏1≤k≤n (z - ζnky)
Since x∈ℤ but most of the factors on the right hand side aren't, there would be a clear violation of unique factorization. Unfortunately, such a violation can't be ruled out, so the proof doesn't work. (It does work for those n where ℚ(ζn) has unique factorization. It wasn't known until 1976 that there are only 29 distinct cyclotomic fields that do have unique factorization.)

Interestingly enough, at almost exactly the same time, Eduard Kummer (1810-93), working independently on questions involving cyclotomic fields, had not only understood the problem of (lack of) unique factorization, but had even started to develop a way around the problem – what he called the method of "ideal numbers" or "divisors". Kummer had also found examples where unique factorization failed in ℤ[ζ37]. He wrote a letter to the mathematicians in Paris who were debating Lamé's work, and pretty much put an end to their deliberations.

Although Kummer's work was not solely concerned with Fermat's Last Theorem, he made what were some of the most significant partial solutions to the problem, and in the process played a huge role in advancing algebraic number theory. His work also led to the theory of ideals as discussed here. Much of what Kummer tried to do was to find "ideal" numbers, of some sort, for which unique factorization could be proven, so that as above a contradiction would arise if Fermat's equation had a solution.

In upcoming installments we'll work with somewhat more abstract ring theory, and eventually find that in any ring of algebraic integers (or in certain rings that are defined more abstractly), there is unique factorization of ideals into prime ideals.

Tags: ,

No comments:

Post a Comment

 

Knowledge Link Suggest

Drug Abuse Online Lottery Games Drug Abuse Health and Fitness Health and Fitness free music downloads Law Reports Wallpapers Artist online shop originals Product Auto Parts Web Developer online shop originals Product Insurance Company Health Life Technology System Home Security Beauty Cosmetic Online Travel Agency Linux Tips Guide Shooting Simulator Gadgets Technology Project Build Houses Payday Loans Health News Information Animals Shop Travel Agency North Owl Network Dentist Healthy Aging asian pretty The Tool Hospital Barbecook Best way to make Money with Forex Trading Green Home Design Online Lottery atozbythesea.eu.org augmentezvosventes.eu.org automobielbedrijf.eu.org aworldofwindows.eu.org aws-orgware.eu.org azregionalmls.eu.org bacimamma.eu.org Business Plan Competition travel channel and rental cars Pharmacy and drugs Criminal Justice Planning Agency Lawyers Association The Best of Insurance http://celebritytidings.eu.org http://livingandfashion.eu.org Academic Advising Audio logo Stock Audio auto repair advice guide Healthy Women Medical Centre busy with business Car Finance Tools Debt Settlement Lawyer crafty stash corner cuentas and colors Cheap Wedding Dresses drug abuse statistics Lifestyle Home Solutions education essay titles finance insurance premium Lawyer and Legal Information guide healthy recipes health problems of children health topics kids home decorating service home improvement project home improvement project management home improvement project planning home insurance morgages house insurance plan house insurance progressive The Savings Bank insurance agency business loans insurance business loans insurance premium loan legacy landscape blocks legacy landscape contractors legacy landscape materials personal insurance loan real estate for commercial Bedroom Furniture Lawyer Requirements Concept Design teach about alcohol abuse technology support autocar the family deal tile flooring ideas tips lifetime fitness tips reduce car insurance quotes Online Jewelry Stores traveller hotel Insurance and Finance Auto and Parts wood flooring ideas the networks board game Adult Learning Theories sexy lingerie tattoo designs wallpaper artist health category insurance financial category Category Online Money Category Home Improvement online business category Category software games and android category marketing advertising category marketing advertising Home Solution Real Estate Financing Lucepedia Sports and Fitness Train Now Business Economy Global auto car loans car loans insurance Health Information how to prevent diabetes Best Credit Cards Cheap Wedding Dresses Small Business Finance Systems Body Health Home Improvement Services Guide Android Apps and Games fashion design Alcoholism Android Phones Board Games Cheap Tickets and Vacation Packages Online Prevent Child Obesity Prevent Premature Ejaculation Labor Lawyers Home Interior Design Ideas plumbers and carpet cleaning service diet and slimming Beautiful Picture Doctors Dental Care music video art BurningPedia Guide Planning Home Improvement Beauty Cosmetic Web Development Company lawyer requirements Books On Children Education security innovation work iPhone and iPad Turkey The Medical School Barbecue Electric The Linux Documentation Project Mental Health Care New Car Buying Guide House Curtains Dentist All System Education iPhone Specifications Cloud Trading Strategy mortgages and loans Pharmaceutical Info Online Application Service Provider The Masterchef Business Service Delivery Arts and Entertainment Pokerstars Real Money The Masterchef Electric Motors Gadgets Technology Casino Zone Small Business Solutions Home Advice Fashion Design Clothing Adventure Games Health and Fitness Hospital Equipment Plan Life Insurance public transportation services Prevent Premature Ejaculation Best Professional Carpet Cleaning Web Hosting Vehicle Emissions Inspection Drug Abuse Collection Travel Agency Medical Herbs color vision casting Aromatherapy Tips community film Cardio Tips Top Best Home Improvements Article Marketing Colon Cleanse Tips Terms and Conditions of Sale Products for Treating Acne Scars Natural Beauty Products Seo Web Marketing Business Prospects Trade Online Tool House Cleaner Bathing Equipment Financial Solutions Credit Loans Business Insurance Best Supplements Online Affiliate Marketing Web Design Jobs Online Business Degrees Travel and Leisure Hotel Business Online Payment Processor Women Management Criminal Law Marketing Product Webdesign Transportation Services The Marketing Business Prospects Trade Online Shop bed furniture free movies online Direct Hardware Supply Free Download PDF Technology Ipad doctors Specials Medical Dictionary Legacy Landscape Bank South Pacific Carpet Cleaner Best Internet Business Seo Web Marketing Doctor Healthy Doctors Diet Program Health Benefits of Drinking Water ups pills drugs Best Supplements dynasty fine jewelry Ceramic Floor Tile Online Business Degrees Property Residential and Commercial Real Estate All About Medical Bankruptcy Prevent Child Obesity Handbag and Shoes Education of Children physiotherapy school Cheap Airline Tickets Healthy Life PET online Interzoo Rehabilitation Center electric hybrid vehicles Baidu Software Center Automotive Advertising Agencies Health and Care Virtual private network Student Education Technosound Wedding Dresses excel archives Classic Car Restoration Defense Audit Banking Travel and holiday Forex Trading Jobs Cheap Car Insurance Opening the Book Safe Chemical Act Presspeople System Software Plan Life Insurance Top Best Home Improvements Cheap Car Insurance Auto Parts and Service Swiss Business Acting in Real Time Fitness Equipment Tips Bodybuilding Tips Search Engine Optimization Cardio Tips Beauty Salon Express The Best Advice Lawyer Advice Associated With Online Payday Spa and Wellness Center Packages Holiday Travel European Countries Tours Small Business Finance Electronic Cigarette Commercial Law Advice on Travel Planning Home and Family All System Education Graphic Visual Photo business between countries Shop Online Financial Security Authority Latest Fashion Trends Photoshop Wallpaper Real Estate Agents Micro Cogeneration Beauty Products Business Travel Insurance International Trucks For Sale Hair Tonic Top Web Design Business Financial Touring Start Vehicle Emissions Inspection Business Service Delivery Insurance and Finance Web Design Business Shopping Online Insurance Processing Systems Electric System Vehicle Blood Medicine European Vehicle Market Pharmaceutical Section Home sale Beauty Therapy Acupuncture Recipes Kitchen Restaurant Hardwood Flooring Resident Services The Masterchef Employment and the Economy Free Web Design Paint Creations Travel Insurance SEO Consulting Services Car Insurance Loans Doctors Dental Care Vehicle Wash Systems Search Engine Marketing Online Education Schools Natural Beauty Site Cheap Wedding Dresses annuaire automobile episiotomy and pregnant Westerly Lawyers Healthcare Travel Vacations Cheap iPhone Specifications The ReaL Estate Event Online Pharmacy and Drugstore Music Video Art CareCredit Cloud Trading Strategy House Cleaning Services Interoperability the talent agency Good Home Design shooting simulator Independent Port Services The Hotel Sax The Digital Money Booking Tickets and Hotels Online Business Loans Department of Pharmacy Law Free Credit Report football development Trending in Entertainment Prison Education Sexy Lingerie Wood Flooring Ideas Dealer Optics and Ophthalmology Car Insurance Online Urbanism and Architecture Electronic money International Conference on Education Dental Care Auto Repair Shop Auto Repair Advice Guide Dental Clinic Online Pharmacy Debt Consolidation Loans Free Software Smartphone Advice on benefits Insurance Providence Real Estate Product Research Analyst Product Research Facilitator Fashion Magazine Substance Abuse Education And Technology Marketing Strategy The Best Payday Advance Incident Report Criminal Accounting Books Caring for Pets Real Estate Financing Banking Software Solutions Financial Solutions Credit Loans Full Service Research Agency Discover our free games Business Prospects Trade Online Restaurant Hood Cleaning how to start a business Women's Health and Beauty Payday Loans Criminal Attorney Lawyers Travel and Tourism Fashion Designer The Food Products Fitness and Sports The Beginner's Guide to Businesses Tips To Make SEO Techniques Online Payday Loans Medical Advocate Englisch Training Online Drivers Education Auto Parts and Service Best hotels cheap sport shoes Dental Work Insurance Company Android Phones Financial Advice Service Mobile phone Free Credit Report  The Hair Tonic Advantage Car Rental Information Civil Law Information Payday Cash Loans Payday Cash Loans Advance Affordable Auto Insurance Effective Quality Services Car seats Product Interactions Specialist Medicina Natural Traveller and Hotel European Investment Bank Vehicle Electrical System Pet Food Shop Online Lawyer Employment Plaza Ballroom Weddings Art School The best form of insurance solutions Advice on benefits Insurance Medical coding Tutor Buying Safe Products Free Online Game Famous Fashion Designers Education in Music Virtual Currency Systems Learn Trading Folktronica music Digital currency Manufacturing Engineering Technology Fashionista The Money System Explained Good Marketing In Business Technology Ipad Real Estate Bubble CarePass for Android Business Direct Marketing Carpet Cleaner Natural Beauty Products Seo Web Marketing Business Prospects Trade Online Tool House Cleaner Bathing Equipment Financial Solutions Credit Loans Business Insurance Best Supplements Online Affiliate Marketing Web Design Jobs Online Business Degrees Travel and Leisure Hotel Property Residential and Commercial Real Estate All About Medical Bankruptcy Prevent Child Obesity Handbag and Shoes Education of Children Women Management Free Online Slots Games Healthy Life PET online Interzoo Sunrise Health and Rehabilitation Center Presspeople System Software Baidu Software Center Scientific Entrepreneurship Automotive Advertising Agencies Health and Care Virtual private network Student Education Technosound Information network energy sources Niche Dating Classic Car Restoration Defense Audit Banking Travel and holiday Safe Chemical Act Leader in internet information Music Electric The Marketing verb translation country island FILM LITERATURE GOLD STATUS A Short History of Movies Sexy Lingerie Classified ads Inspire Home Design Trauma Clinic Finance Investment Business Marketing Acting in Real Time Glasgow Tourism Herbal Medicine Cheap Car Insurance Quotes Health Pregnancy Professional Graphic and New Media Designers Legal Connect Network Generating Money Online Oregon health plan Slow Cooker Recipes Best Internet Business Terms and Conditions of Sale Products for Treating Acne Scars Hardwoods Furniture Green Tea Fitness Equipment Tips Bodybuilding Tips Forex Trading Article Marketing Colon Cleanse Tips Hair Loss Tips List Building Search Engine Optimization Cardio Tips Aromatherapy Tips Medical Weight Loss Clinic The Benefits Of Moving Insurance Travel Advice Online Dating Security Network Information Green Energy Scheme Group Auto International Business Telecommunication International All Medicine Reviews Marketplace Real Estate Review Clothing Style Management Consulting Real estate Natural Medicine Treatment Natural Beauty Skin Care Products Shop online Marketing Bank rent car guide Fishing Games cheap website traffic Health and Beauty Gambling football Clothes and Accessories Garden and Landscaping Portal Game Online E-commerce Business Car Accident Lawyer Criminal Law Best free Android apps Hotel and Travel Jobs Technology Film Industry Free Music Downloads Solar Installation Search Engine Optimization Life Technologies Football Betting Site Substance Abuse Apartments home improvement contractor job Luxury Car News Law Products and Services Business Web Hosting Company Complaints Diabetes Drug Abuse Marketing Product Make Money Online Living Without Money Apartments Luxury Car News Law Products and Services Growth Technologies Articles on business growth technologies information Health Insurance Management Advisors System Solar Panel Online Trading The Best of Insurance Payday Loans for Bad Credit Financial Solutions Credit Loans Ensure Insurance Plan Financial Management Team Payday Loans Reviews Distributed File System Student Loan Payments Cheap Insurance in Los Angeles Favorite Fashion Styles The Best Savings Rates Manage Student Loans Webshop europe business guide Shop Personal Loans debt consolidation loans Personal Loan Repayment Chiropractic Tables Webdesign Management Concepts Android Software Free Internet Solution the online casino Best Small Business Blogs Sound Beginnings Automotive Electrical Systems Skin Care in Centrum Master Ethique Cher Client Agency Law Sell Products Online Mens Shoes Strategy Business Development Women's shoes Headache Causes Shop Online Clothing Online Pharmacy Healthy Aging Battle Games best electric vehicle Commercial Real Estate Cosmetic Beauty Product Health Medicine Pharmacy Image of the Orient Fashion Jewellery

Blog Archive