A classification of the extensions of degree \(p^2\) over \(\mathbb Q_p\) whose normal closure is a \(p\)-extension
type: Item
instance of: scholarly article
title: A classification of the extensions of degree \(p^2\) over \(\mathbb Q_p\) whose normal closure is a \(p\)-extension
zbMATH Open document ID: 1161.11034
author: Luca Caputo
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/10800
full work available at URL: article02.pdf
review text: The main content of this paper is a classification theorem for degree \(p^2\) extensions of \(\mathbb{Q}_p\) having normal closure a \(p\)-extension. Call the set of such extensions \(\mathcal{E}_p\). The author first shows (Prop. 2.1) that \(L\in\mathcal{E}_p\) if and only if \(L|\mathbb{Q}_p\) contains an intermediate field \(K\) such that both \(L|K\) and \(K|\mathbb{Q}_p\) are \(p\)-cyclic. Since the \(p\)-cyclic extensions of \(\mathbb{Q}_p\) are known by local class field theory, it suffices to give a description of the set \(\mathcal{E}_p(K)=\{L\in\mathcal{E}_p \;| \;K\subset L\}\), for each \(p\)-cyclic extension \(K|\mathbb{Q}_p\). Set \(M_p(K)\) to be the compositum of the extensions in \(\mathcal{E}_p(K)\). Then \(M_p(K)|K\) is the maximal \(p\)-elementary abelian extension of \(K\). Moreover, \(M_p(K)|\mathbb{Q}_p\) is Galois, and the isomorphism type of the group \(G_p(K)=\text{Gal}(M_p(K)|\mathbb{Q}_p)\) is independent of the chosen \(p\)-cyclic extension \(K|\mathbb{Q}_p\) (Prop. 2.2). The author obtains the classification theorem for \(\mathcal{E}_p(K)\) by making a careful study of the group \(G_p(K)\). Proposition 3.1 gives an explicit presentation for \(G_p(K)\) as a p-group with \(p+1\) generators. Letting \(H_p(K)=\text{Gal}(M_p(K)|K)\) denote the maximal \(p\)-elementary abelian subgroup of \(G_p(K)\), the author goes on the classify the normal subgroups of \(G_p(K)\) that are contained in \(H_p(K)\) (Prop. 4.1). By the Galois correspondence, the normal closures of the extensions in \(\mathcal{E}_p(K)\) are the fixed fields of a subset of these normal subgroups. The classification theorem for \(\mathcal{E}_p(K)\) is obtained by identifying this subset explicitly (Section 5). In the remainder of the paper, the author determines the ramification filtrations and differents for the normal closures of the elements of \(\mathcal{E}_p\). All of the results are nicely summarized in a table at the end of the paper. The author mentions in the introductio…
reviewed by: Scott Corry
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
zbMATH DE Number: 5302778
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: \(p\)-extension
zbMATH Keywords: Galois group
MaRDI profile type: Publication
OpenAlex ID: W2317303779
cites work: Q4544915
cites work: Q5656854
cites work: On 𝑝-extensions
cites work: Q3856855
cites work: Q5800826
DOI: 10.5802/JTNB.590
description: scientific article; zbMATH DE number 5302778