×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] M.J. Greenberg, An elementary proof of the Kronecker-Weber theorem. Amer. Math. Monthly 81 (1974), 601-607; corr.: ibid. 82 (1975), 803 · Zbl 0307.12012
[2] D. Hilbert, Ein neuer Beweis des Kronecker’schen Fundamentalsatzes über Abel’sche Zahlkörper. Gött. Nachr. (1896), 29-39 · JFM 27.0062.03
[3] D. Hilbert, Die Theorie der algebraischen Zahlkörper. Jahresber. DMV 1897, 175-546; Gesammelte Abh. I, 63-363; Engl. Transl. by I. Adamson, Springer-Verlag 1998
[4] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory. Springer Verlag 1982; 2nd ed. 1990 · Zbl 0482.10001
[5] F. Lemmermeyer, Reciprocity Laws. From Euler to Eisenstein. Springer Verlag 2000 · Zbl 0949.11002
[6] D. Marcus, Number Fields. Springer-Verlag 1977 · Zbl 0383.12001
[7] A. Speiser, Die Zerlegungsgruppe. J. Reine Angew. Math. 149 (1919), 174-188
[8] E. Steinbacher, Abelsche Körper als Kreisteilungskörper. J. Reine Angew. Math. 139 (1910), 85-100
[9] L. Washington, Introduction to Cyclotomic Fields. Springer-Verlag 1982 · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.