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The number of semidihedral or modular extensions of a local field. (English) Zbl 1154.11042

For \(k\) a field and \(G\) a finite group, let \(\nu(k,G)\) denote the number of Galois extensions of \(k\) (up to isomorphism) having Galois group \(G\). This article calculates \(\nu(k,SD_{2^m})\) and \(\nu(k,M_{2^m})\) for \(k\) a local field, i.e. a finite extension of the \(p\)-adic field \(\mathbb{Q}_p\), \(p\) a prime, and where \(SD_{2^m}\) denotes the semidihedral group of order \(2^m\) and \(M_{2^m}\) denotes the modular group of order \(2^m\). This completes the calculation of \(\nu(k,G)\) over local fields \(k\) for groups \(G\) of order \(2^m\) and having elements of order \(2^{m-1}\), as the second author has previously calculated \(\nu(k,D_{2^m}\)) and \(\nu(k,Q_{2^m})\) for \(k\) a local field, where \(D_{2^m}\) and \(Q_{2^m}\) denote the dihedral and generalized quaternion groups of order \(2^m\), respectively [see M. Yamagishi, Proc. Am. Math. Soc. 123, No. 8, 2373–2380 (1995; Zbl 0830.11045)].
The result in the tame case (when the characteristic of the residue field of \(k\) is not 2) is already more or less well known [see C. U. Jensen, Finite groups as Galois groups over arbitrary fields, Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 2, 435–448 (1992; Zbl 0780.12004)], but is provided in the article.
The proof in the wild case (when the characteristic of the residue field of \(k\) is 2) uses a general formula for \(\nu(k,G)\) for \(k\) a local field and \(G\) a \(p\)-group (\(p\) a prime), obtained previously in the article cited above by the second author.

MSC:

11S20 Galois theory
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References:

[1] G. Fardoux, Sur les extensions pseudodiédrales d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A145- A148. · Zbl 0264.12005
[2] M. Ito, On \(2\)-extensions of a local field, Master’s thesis, Nagoya Institute of Technology, 2007. (in Japanese).
[3] C. U. Jensen, Finite groups as Galois groups over arbitrary fields, in Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), 435-448, Contemp. Math., Part 2, Amer. Math. Soc., Providence, RI, 1992. · Zbl 0780.12004
[4] M. Yamagishi, On the number of Galois \(p\)-extensions of a local field, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2373-2380. · Zbl 0830.11045 · doi:10.2307/2161262
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