A classification of the extensions of degree p 2 over p whose normal closure is a p-extension
Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 337-355.

Soit k une extension finie de p et soit k l’ensemble des extensions de degré p 2 sur k dont la clôture normale est une p-extension. Pour chaque discriminant fixé, nous calculons le nombre d’éléments de p qui ont un tel discriminant, et nous donnons les discriminants et les groupes de Galois (avec leur filtrations des groupes de ramification) de leurs clôtures normales. Nous montrons aussi que l’on peut généraliser cette méthode pour obtenir une classification des extensions qui appartiennent à k .

Let k be a finite extension of p and 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 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 k .

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     author = {Caputo, Luca},
     title = {A classification of the extensions of degree $p^{2}$ over $\mathbb{Q}_{p}$ whose normal closure is a $p$-extension},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {337--355},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
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Caputo, Luca. 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, Tome 19 (2007) no. 2, pp. 337-355. doi : 10.5802/jtnb.590. http://www.numdam.org/articles/10.5802/jtnb.590/

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