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?

 A: 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.
A: 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.
A: 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.
