# 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?

-
You should ask Norbert Schappacher (see www-irma.u-strasbg.fr/~schappa/NSch/Home.html ). –  Damian Rössler Aug 30 '11 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 '11 at 16:15
Here is Hilbert's 1896 paper digizeitschriften.de/dms/img/… –  Gjergji Zaimi Aug 30 '11 at 16:21
See section 2 of emis.de/journals/SC/1998/3/pdf/smf_sem-cong_3_243-273.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 '11 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.

-

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.

-
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 '11 at 16:39
(I fixed your Spanish :) ) –  Mariano Suárez-Alvarez Aug 30 '11 at 16:40
Thank you Mariano! I can read Spanish, but I write it very badly :) –  Francesco Polizzi Aug 30 '11 at 16:41