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

Yamagishi, Masakazu

doi:10.1090/S0002-9939-1995-1264832-0

On the number of Galois $p$-extensions of a local field
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Proc. Amer. Math. Soc. 123 (1995), 2373-2380 Request permission

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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