Ramble, Part 50: Idealism
May. 21st, 2008 02:52 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe ideas described in the previous Ramble, concerning integer solutions to the equation a2+b2=c2, inspired the French mathematician Gabriel Lamé to an attempt at proving Fermat's Last Theorem. Lamé was no stranger to the problem; he had proven that the equation a7+b7=c7 had no integer solutions except the trivial ones where one of a, b, c is 0 - a special case of Fermat's Last. Lamé presented a solution to the entire theorem. His solution was quickly shown to be flawed, but attempts to repair the flaw proved fruitful.
Lamé's idea was this. Consider the equation an+bn=cn, where n is an odd prime greater than 1. (It's fairly easy to show that, if Fermat's Last holds for n=4 and for odd prime values of n, then it holds for all n greater than 2.) Let z=cos(2 pi/n)+i sin(2 pi/n). z has the property zn=1, and every complex number with that property is a power of z. Now, an+bn=(a+b)(a+bz)(a+bz2)...(a+bzn-1). These factors are relatively prime in pairs; that is, no two of them have any common factors. If this product is to equal cn, then, said Lamé, each of the factors must be an nth power (possibly multiplied by a unit). He then showed that this would imply the existence of a smaller triplet a, b, c with the same property; invoking Fermat's Method of Infinite Descent, he declared the theorem proven.
There was one flaw in his proof. Just as Gauss's construction took place in the ring of Gaussian integers, Z[i], so Lamé's took place in a ring Z[z], consisting of complex numbers of the form m+nz+oz2+...+pzn-1, where m, n, o, ... p are integers. Now, the argument about nth powers works in the integers (and in the Gaussian integers) because of the Unique Factorization Theorem, that every integer (or Gaussian integer) can be written as a product of primes in essentially only one way. But what guarantee do we have that it holds in Z[z]?
The point of all this is that it had become clear that the proper study of algebra - or, at least, its most important sub-branch - was precisely the study of rings, of sets in which operations behaving like addition and multiplication could be defined. It was no longer sufficient to say "number"; what kind of number had to be specified, and each kind of number had to be studied more-or-less autonomously. Thus, from one point of view, algebra had splintered; from another, its field of action had expanded dramatically. Just as had happened with geometry, this new range of possibilities was to prove valuable.
Previous Next
Ramble Contents
Lamé's idea was this. Consider the equation an+bn=cn, where n is an odd prime greater than 1. (It's fairly easy to show that, if Fermat's Last holds for n=4 and for odd prime values of n, then it holds for all n greater than 2.) Let z=cos(2 pi/n)+i sin(2 pi/n). z has the property zn=1, and every complex number with that property is a power of z. Now, an+bn=(a+b)(a+bz)(a+bz2)...(a+bzn-1). These factors are relatively prime in pairs; that is, no two of them have any common factors. If this product is to equal cn, then, said Lamé, each of the factors must be an nth power (possibly multiplied by a unit). He then showed that this would imply the existence of a smaller triplet a, b, c with the same property; invoking Fermat's Method of Infinite Descent, he declared the theorem proven.
There was one flaw in his proof. Just as Gauss's construction took place in the ring of Gaussian integers, Z[i], so Lamé's took place in a ring Z[z], consisting of complex numbers of the form m+nz+oz2+...+pzn-1, where m, n, o, ... p are integers. Now, the argument about nth powers works in the integers (and in the Gaussian integers) because of the Unique Factorization Theorem, that every integer (or Gaussian integer) can be written as a product of primes in essentially only one way. But what guarantee do we have that it holds in Z[z]?
Let me give an example. The ring Z[Sqrt[-5]] is not one of the rings Z[z], but it will serve to illustrate the point. It consists of the complex numbers m+n Sqrt[-5], where, as usual, m, n are integers. The number 9 can be factored as 3x3 or as (2+Sqrt[-5])x(2-Sqrt[-5]); the factorization is not unique. (Technically, these are not factorizations into primes but into irreducibles; I won't bother you with the details of the distinction, except to say that in the integers primes and irreducibles are the same thing, and that for these purposes it is factorization into irreducibles that is important.)In an attempt to retrieve the situation, the German mathematician Ernst Kummer developed what he called "ideal numbers" - the jargon today is simply "ideals"; it turns out that (in this context) ideals do factor uniquely, but this is not enough to rescue Lamé's proof. However, two major lessons were learned. First, different sets of numbers - integers, Gaussian integers, Z[z], and the like - are worth studying, each in its own right, and they behave differently. Some of them had the unique factorization property; others do not. Some of them have only a few units; some have many, even infinitely many. They have many properties in common, and the study of those properties has been fruitful, but they also vary considerably in other ways, and the modern field of algebraic number theory is devoted to their study. The second lesson was that Kummer's "ideals" are a valuable tool in that study, and - by a long and winding road - they turned out to be key to the eventual proof of Fermat's Last Theorem (and much else besides).
The point of all this is that it had become clear that the proper study of algebra - or, at least, its most important sub-branch - was precisely the study of rings, of sets in which operations behaving like addition and multiplication could be defined. It was no longer sufficient to say "number"; what kind of number had to be specified, and each kind of number had to be studied more-or-less autonomously. Thus, from one point of view, algebra had splintered; from another, its field of action had expanded dramatically. Just as had happened with geometry, this new range of possibilities was to prove valuable.
Previous Next
Ramble Contents
