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On the number of Galois $ p$-extensions of a local field

Author: Masakazu Yamagishi
Journal: Proc. Amer. Math. Soc. 123 (1995), 2373-2380
MSC: Primary 11S20; Secondary 11S15
MathSciNet review: 1264832
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Abstract: Let p be a prime, k a finite extension of the p-adic field $ {\mathbb{Q}_p}$, and G a finite p-group. Let $ v(k,G)$ denote the number of non-isomorphic Galois extensions of k whose Galois groups are isomorphic to G. When k does not contain a primitive p-th root of unity, I. R. Šafarevič gave an explicit formula for $ v(k,G)$. In this note, we treat the case when k contains a primitive p-th root of unity. After giving a general formula for $ v(k,G)$ (Theorem 1), we calculate $ v(k,G)$ explicitly for some special p-groups (Theorem 2.2).

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Keywords: Local field, Demuškin group, p-extension
Article copyright: © Copyright 1995 American Mathematical Society

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