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](http://en.wikipedia.org/wiki/Class_field_theory#Formulation_in_contemporary_language) even states that Kronecker *conjectured* the result!)



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> When was the theorem *finally* proved, exactly?
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The first complete proof of the Kronecker-Weber theorem
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.


