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On the number of Galois \(p\)-extensions of a local field. (English) Zbl 0830.11045

By a well known result of Krasner, a \(p\)-adic field \(k\) has only a finite number of non-isomorphic algebraic extensions with same fixed degree. In this paper, the author calculates the number \(\nu (k, G)\) of Galois extensions over \(k\) with prescribed finite Galois group \(G\), mainly a \(p\)-group. His method uses the structure of the Galois group \({\mathcal G}\) of the maximal pro-\(p\)-extension over \(k\). When \(k\) contains the group \(\mu_p\) of \(p\)-th roots of unity, a general algebraic formula giving \(\nu (k, G)\) is obtained via the classification theorem of Demushkin groups. Some explicit computations are provided for special \(p\)-groups which generalize several previous results. When \(\mu_p \not\subset k\), the author recovers Shafarevich’s classical formula in the case where \({\mathcal G}\) is a free pro-\(p\)-group.

MSC:

11S20 Galois theory
11R32 Galois theory
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