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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 Wenber'sWeber's.
show/hide this revision's text 1

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 Wenber's.