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?
Best Answer
The correct reference is
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.