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Luca Caputo
A classification of the extensions of degree $p^{2}$ over $\mathbb{Q}_{p}$ whose normal closure is a $p$-extension
Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 337-355, doi: 10.5802/jtnb.590
Article PDF | Reviews MR 2394890 | Zbl 1161.11034 | 1 citation in Cedram

Résumé - Abstract

Let $k$ be a finite extension of $\mathbb{Q}_{p}$ and $\mathcal{E}_{k}$ be the set of the extensions of degree $p^{2}$ over $k$ whose normal closure is a $p$-extension. For a fixed discriminant, we show how many extensions there are in $\mathcal{E}_{\mathbb{Q}_{p}}$ with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in $\mathcal{E}_{k}$.

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