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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.
##### MSC:
 11R18 Cyclotomic extensions
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##### References:
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