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.