A classification of the extensions of degree \(p^2\) over \(\mathbb Q_p\) whose normal closure is a \(p\)-extension. (English) Zbl 1161.11034

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 introduction that it would be easy to generalize the results of this paper to the case where \(\mathbb{Q}_p\) is replaced by an arbitrary finite extension \(k|\mathbb{Q}_p\). Indeed, the first 3 sections of the paper are set in the context of an arbitrary \(p\)-adic ground field \(k\), although this reviewer has only stated the results for \(k=\mathbb{Q}_p\). The author restricts to \(\mathbb{Q}_p\) starting with Proposition 4.1, the classification of the normal subgroups of \(G_p(K)\).


11S15 Ramification and extension theory
11S20 Galois theory


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