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Kronecker-Weber via Stickelberger. (English) Zbl 1103.11030
The theorem of Kronecker-Weber classifying the abelian extensions of \(\mathbb Q\) as being exactly the cyclotomic ones, has numerous proofs. These are often a posteriori and exemplify the consequences of some theory.
In this brief and interesting paper, Lemmermeyer shows that the theorem can be reducded to a fact which he then proves using Stickelberger’s theorem. The reduction is probably of older date and the author traces it back to a paper by F. Steinbacher in 1910 [J. Reine Angew. Math. 139, 85–100 (1910; JFM 41.0246.02)]; but the Stickelberger argument is new and elegant.
If \(p\) is a prime, then the maximal abelian extension \(\mathbb K_p / \mathbb Q\) which has exponent \(p\) and is unramified outside \(p\) is cyclic, being the subfield of degree \(p\) in \(\mathbb Q(\zeta_{p^2})\). The proof is prepared by Kummer theory and adjoining a \(p\)-th root of unity: \(\mathbb L = \mathbb K[ \zeta_p ],\) for some cyclic subfield \(\mathbb K \subset \mathbb K_p\) and obtaining abelian root extensions of \(\mathbb Q(\zeta_p)\). By invoking the Stickelberger Theorem and investigating the possible decompositions of \(\alpha \in \mathcal{O}(\mathbb Q(\zeta))\) such that \(\mathbb L = \mathbb Q(\zeta, \alpha^{1/p})\), Lemmermeyer shows that \(\alpha\) must be a unit.
11R18 Cyclotomic extensions
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