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

finallyproved, exactly?