## The first complete proof of the Kronecker-Weber theorem

While the Kronecker-Weber theorem —that every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field— is always attributed to, well, Leopold Kronecker and Heinrich Martin Weber, most sources I've seen that care to go into such details observe that their proofs were incomplete and were later fixed by others, among which one usually finds Hilbert named (One extreme example: Wikipedia even states that Kronecker conjectured the result!)

When was the theorem finally proved, exactly?

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You should ask Norbert Schappacher (see www-irma.u-strasbg.fr/~schappa/NSch/Home.html ). – Damian Rössler Aug 30 2011 at 16:12
You already mention that Hilbert is given recognition as having had the first proof (1896). Why not look into that some more? Washington's book on cyclotomic fields gives a proof in Chapter 14 and he refers the reader to a paper of O. Neumann from Crelle 323 (1981), 105--126 which might give some information about what exactly was in or not in the arguments of Kronecker and Weber. – KConrad Aug 30 2011 at 16:15
Here is Hilbert's 1896 paper digizeitschriften.de/dms/img/… – Gjergji Zaimi Aug 30 2011 at 16:21
See section 2 of emis.de/journals/SC/1998/3/pdf/… for Schappacher's rather detailed historical survey. His conclusion is that the first complete proof is due to Hilbert. – Denis Chaperon de Lauzières Aug 30 2011 at 16:25

The correct reference is

• Olaf Neumann, Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber", J. Reine Angew. Math. 323 (1981), 105-126

This is also the source that Schappacher relies on. Neumann analyses Weber's first proofs (there's not much of a proof in Kronecker) and points out his errors (he overlooked that the Galois group does not always act nicely on Lagrange resolvents if the fields in question have a nonempty intersection). Weber's proofs, strictly speaking, were only fixed by Neumann; the proofs in between did not use Lagrange resolvents, except for a proof by Mertens which suffers from the same defects as Weber's.

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I'm just reading the interesting book by Jeremy J. Gray "The Hilbert challenge" (well, I actually have its spanish translation "El reto de Hilbert").

In Chapter 3, describing the 12th Hilbert's problem, Gray says that the first correct proof was given by Hilbert. In fact, quoting my book:

"La cuestión de lo que se puede atribuir a Kronecker a modo de demostración es bastante difícil, y también es falsa la sugerencia de que la primera demostración válida fue dada por Weber (el error de Weber no fue detectado hasta 1979). De hecho, parece que fue el propio Hilbert el primero en demostrar el teorema de Kronecker-Weber".

The reference given is Schappacher's paper "On the history of Hilbert's Twelfth Problem" published by the Societé Mathematique de France (1998).

EDIT. After I finished to write this answer I read the comment by Denis Chaperon de Lauzières, saying the same thing.

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Free translation back into English (!): «The question of what of the proof can be attributed to Kronecker is very difficult, and it is also not true that the first proof is due to Weber (Weber's mistake was not found until 1979) In fact, it seems that Hilbert himself was the first to prove the Kronecker-Weber theorem.» – Mariano Suárez-Alvarez Aug 30 2011 at 16:39
(I fixed your Spanish :) ) – Mariano Suárez-Alvarez Aug 30 2011 at 16:40
Thank you Mariano! I can read Spanish, but I write it very badly :) – Francesco Polizzi Aug 30 2011 at 16:41

Weber gave the first complete proof, based partly on ideas of Kronecker. It's true that there are errors in Weber's proofs, but nothing that he couldn't have fixed if they had been pointed out to him. Kronecker and Weber had some of the most original and magnificently beautiful ideas in mathematics --- let the lesser mathematicians fuss over the details.

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Among the errors in one of Weber's proof not pointed out by Neumann is his theorem that in cyclic extensions of number fields with prime power degree, a prime that ramifies must ramify completely. Of course he could have circumvented this "result" had someone pointed out to him that it was nonsense. Even Goldbach managed to prove that -1 is not a square modulo primes 4n+3 after Euler had politely corrected about a dozen errors in his attempts to do so. And as far as I can see, no one so far has disputed the depth of Kronecker's (and, perhaps to a lesser degree, Weber's) ideas. – Franz Lemmermeyer Aug 31 2011 at 12:39
Well, they are questioning whether Weber proved the theorem, and even whether it should be called the Kronecker-Weber theorem. If you look hard enough, you can find errors in a great many proofs: the giants prove the great theorems; the pygmies fix their proofs. – anon Aug 31 2011 at 15:14
Well, sometimes the giants do not prove the great theorems. There's a paper of Kronecker in which he "proves" that primes splitting completely in a normal extension K/Q have density 1/(K:Q). Actually he doesn't prove anything. Even worse, I am not able to tell whether some of his statements are conjectures or whether he actually thought he had a proof. I would certainly vote it to be one of the best papers he wrote, but filling the gaps, at least in this case, is not for pygmies, neither early nor late in the day. – Franz Lemmermeyer Aug 31 2011 at 17:48
We are talking about Weber's proof, not Kronecker's. I'm strongly objecting to Gray's statement "it is also not true that the first proof is due to Weber" (and also to the sort of mentality that leads to such statements). I also object to his statement that "Weber's mistake was not found until 1979". How would he know? Not everyone writes a paper every time they find a mistake. – anon Aug 31 2011 at 19:31
I agree with both of your objections. Weber's proof was a proof, but it had a gap, which was pointed out to him by Frobenius, whose observation was incorporated into the corrigenda of Weber's third proof. BTW, at least in the above quote in Spanish, Gray talks not about the "first proof", but about the "first valid proof". – Franz Lemmermeyer Sep 6 2011 at 11:15