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The number of modular extensions of odd degree of a local field. (English) Zbl 1225.11154

Summary: The number of Galois extensions, up to isomorphism, of a local field whose Galois groups are isomorphic to the modular group \(M_{p^{m}}=\langle x,y \mid x^{p^{m-1}}=y^{p}=1,y^{-1}xy=x^{p^{m-2}+1}\rangle\), where \(p\) is an odd prime, is counted.

MSC:

11S20 Galois theory
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References:

[1] P. Hall, The Eulerian functions of a group, Quart. J. Math. Oxford Ser. 7 (1936), 134-151. · Zbl 0014.10402
[2] M. Ito and M. Yamagishi, The number of semidihedral or modular extensions of a local field, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 2, 10-13. · Zbl 1154.11042 · doi:10.3792/pjaa.83.10
[3] I. Shafarevitch, On \(p\)-extensions, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 351-363.
[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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