Determination of the number of isomorphism classes of extensions of a \kp-adic field

2010/11/01 by Maurizio Monge, Monge, Maurizio · 1 citation
Mathematics · #11S15 #12B25 #20D60 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #advanced mathematical theories #secondary: 05A19

paper · pdf · doi:10.48550/arxiv.1011.0357

openalex publication_date 2010/11/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We deduce a formula enumerating the isomorphism classes of extensions of a \kp-adic field K with given ramification e and inertia f. The formula follows from a simple group-theoretic lemma, plus the Krasner formula and an elementary class field theory computation. It shows that the number of classes only depends on the ramification and inertia of the extensions K/\Q_p, and K(ζ_p^m)/K obtained adding the p^m-th roots of 1, for all p^m dividing e.

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• Families of p-adic fields

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