The first complete proof of the Kronecker-Weber theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:47:18Z http://mathoverflow.net/feeds/question/74073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem The first complete proof of the Kronecker-Weber theorem Mariano Suárez-Alvarez 2011-08-30T15:44:48Z 2011-08-31T11:54:32Z <p>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: <a href="http://en.wikipedia.org/wiki/Class_field_theory#Formulation_in_contemporary_language" rel="nofollow">Wikipedia</a> even states that Kronecker <em>conjectured</em> the result!)</p> <blockquote> <p>When was the theorem <em>finally</em> proved, exactly?</p> </blockquote> http://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem/74081#74081 Answer by Francesco Polizzi for The first complete proof of the Kronecker-Weber theorem Francesco Polizzi 2011-08-30T16:36:20Z 2011-08-30T16:42:35Z <p>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").</p> <p>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:</p> <p>"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".</p> <p>The reference given is Schappacher's paper "On the history of Hilbert's Twelfth Problem" published by the Societé Mathematique de France (1998). </p> <p><strong>EDIT.</strong> After I finished to write this answer I read the comment by Denis Chaperon de Lauzières, saying the same thing.</p> http://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem/74098#74098 Answer by Franz Lemmermeyer for The first complete proof of the Kronecker-Weber theorem Franz Lemmermeyer 2011-08-30T20:04:03Z 2011-08-31T02:02:31Z <p>The correct reference is </p> <ul> <li>Olaf Neumann, <em>Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber"</em>, J. Reine Angew. Math. 323 (1981), 105-126 </li> </ul> <p>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.</p> http://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem/74112#74112 Answer by anon for The first complete proof of the Kronecker-Weber theorem anon 2011-08-31T00:45:16Z 2011-08-31T11:54:32Z <p>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.</p>