What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?

Question

This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields is essentially unfounded, and that Kummer was instead movitated by the desire to formulate and prove general higher reciprocity laws.

My own (not particular well-informed) undestanding is that the problem of higher reciprocity laws was indeed one of Kummer's substantial motivations; after all, this problem is a direct outgrowth of the work of Gauss, Eisenstein, and Jacobi (and others?) in number theory. However, Kummer did also work on FLT, so he must have regarded it to be of some importance (i.e. important enough to work on).

Is there a consensus view on the role of FLT as motivating factor in Kummer's work? Was his work on it an afterthought, something that he saw was possible using all the machinery he had developed to study higher reciprocity laws? Or did he place more importance on it than that? (Am I right in also thinking that there were prizes attached to its solution which could also have played a role in directing his attention to it? If so, did they actually play any such role?)

Answer 1

When Kummer started working on research problems,he tried to solve what became known as "Kummer's problem", i.e., the determination cubic Gauss sums (its cube is easy to compute). Kummer asked Dirichlet to find out whether Jacobi or someone else had already been working on this, and to send him everything written by Jacobi on this subject. Dirichlet organized lecture notes of Jacobi's lectures on number theory from 1836/37 (see also this question), where Jacobi had worked out the quadratic, cubic and biquadratic reciprocity laws using what we now call Gauss and Jacobi sums.

In his first article, Kummer tried to generalize a result due to Jacobi, who had proved (or rather claimed) that primes $5n+1$, $8n+1$, and $12n+1$ split into four primes in the field of 5th, 8th and 12th roots of unity. Kummer's proof of the fact that primes $\ell n+1$ split into $\ell-1$ primes in the field of $\ell$-th roots of unity (with $\ell$ an odd prime) was erroneous, and eventually led to his introduction of ideal numbers.

After Lame (1847) had given is "proof" of FLT in Paris, Liouville had observed that there are gaps related to unique factorization; he then asked his friend Dirichlet in a letter whether he knew that Lame's assumption was valid (Liouville only knew counterexamples for quadratic fields). A few weeks later, Kummer wrote Liouville a letter. In the weeks between, Kummer had looked at FLT and found a proof based on several assumptions, which later turned out to hold for regular primes. Kummer must have looked at FLT before, because in a letter to Kronecker he said that "this time" he quickly found the right approach.

The Paris Prize, as John Stilwell already wrote, did play a role for Kummer, as he confessed in one of his letters to Kronecker that can be found in Kummer's Collected Papers. But mathematically, Kummer attached importance only to the higher reciprocity laws.

Kummer worked out the arithmetic of cyclotomic extensions guided by his desire to find the higher reciprocity laws; notions such as unique factorization into ideal numbers, the ideal class group, units, the Stickelberger relation, Hilbert 90, norm residues and Kummer extensions owe their existence to his work on reciprocity laws. His work on Fermat's Last Theorem is connected to the class number formula and the "plus" class number, and a meticulous investigation of units, in particular Kummer's Lemma, as well as the tools needed for proving it, his differential logarithms, which much later were generalized by Coates and Wiles. Some of the latter topics were helpful to Kummer later when he actually proved his higher reciprocity law.

Here's my article on Jacobi and Kummer's ideal numbers.

Answer 2

Franz Lemmermeyer is better qualified than I am to answer this question, and he certainly knows all about Kummer's motivation from reciprocity laws. However, to answer some parts of your question: Kummer certainly did publish papers on Fermat's last theorem, and he received a prize from the Paris Academy in 1857 for his work on FLT.

He came under consideration for his long paper in the J. de Math. Pures et Appl. of 1851. This was his first extended exposition of his work on cyclotomic integers and ideal prime factorization, and he concluded it by proving a special case of FLT. Thus, even if his real motivation was reciprocity laws, he apparently thought that the best way to "sell" his theory (at least to the French) was by an application to FLT.

Answer 3

I too defer to Franz's deep knowledge on such topics. Everything that I have read in both the primary and secondary literature completely agrees with what Franz has written here and elsewhere. Besides the link I gave to the discussion in his interesting paper on Jacobi and Kummer's ideal numbers, one may also find helpful the following passage from p. 15 of his beautiful book on reciprocity laws. It provides a concise summary of what we currently know about such matters. I quote it below since some readers may not have convenient access to the book.

The role of Fermat's Last Theorem in the development of algebraic number theory is often overrated, probably due to Hensel's (false) story claiming Kummer's first manuscript to have been an incorrect proof of this problem. The crème de la crème of French mathematicians - Lame, Legendre, Liouville, and Cauchy - tried their luck but didn't really advance algebraic number theory during their work on FLT. Gauss did not value it very highly, but admits that it made him take up his investigations in number theory again: in a letter to Olbers from March 21, 1816, he writes

I admit, that Fermat's Theorem, as an isolated result, has little interest for me, since I can easily make a lot of such claims that can be neither proved nor disproved. Nevertheless it made me take up again some old ideas about a large extension of the higher arithmetic. [...] Yet I am convinced, if luck should do more than I may expect and if I succeed in making some of the main steps in that theory, then Fermat's Theorem will appear as one of the less interesting corollaries.

Gauss's last remark clearly indicates that he was at least thinking about the arithmetic in cyclotomic number fields ${\mathbb Q}(\zeta_p)$, even when, in a letter to Bessel a few months later, he reveals that the investigations in question had to do with the part that he eventually would publish, namely the theory of biquadratic residues. Parts of his research were published in 1828 [272] and 1832 [273], and the last paper contains the statement that cubic reciprocity is best described in ${\mathbb Z}[\rho]$, where $\rho^2 + \rho + 1 = 0$, and that, more generally, the study of higher reciprocity laws should be done after adjoining higher roots of unity.

Even Kummer, who is responsible for the greatest advance towards of Fermat's Last Theorem before the recent developments, got the main motivation for studying cyclotomic fields from his desire to find a general reciprocity law (which he called his "main enemy" in [Ku, Feb 25, 1848]). In almost every letter to Kronecker written between 1842 and 1848, Kummer mentions results related to reciprocity; the Fermat equation is mentioned for the first time in [Ku, Apr. 02, 1847]. In [Ku, Sept. 17, 1849] he informs Kronecker about the prize of 3000 Francs that the French Academy had offered to pay for a solution of Fermat's Last Theorem, and in [Ku, Jan. 14, 1850] he writes

Once I will have fathomed whether this is so or not, then I will drop the avaricious plans and work again only for the science, especially for the reciprocity laws for which I have already envisaged some ideas.

It is therefore safe to say that it was the quest for higher reciprocity laws that made Kummer study abelian extensions of $\mathbb Q$, Eisenstein those of ${\mathbb Q}(i)$ and Hilbert those of general number fields. Ironically, it was Hilbert himself who started the rumour that it was FLT that led Kummer to his ideal numbers: in his famous address at the ICM in Paris 1900, he wrote

stimulated by Fermat's problem, Kummer arrived at the introduction of his ideal numbers and discovered the theorem of unique factorization of the integers of cyclotomic fields into ideal prime factors.

Already in 1910, Hensel talked about "incontestable evidence" (actually it was something that Gundelfinger had heard from H.G. Grassmann) for the existence of a manuscript in which Kummer had claimed to have solved Fermat's Last Theorem, a rumour eventually dismissed by Edwards [Ed1, Ed2] and Neumann [Neu].