Let be a finite extension of and be the set of the extensions of degree over whose normal closure is a -extension. For a fixed discriminant, we show how many extensions there are in 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 .
Soit une extension finie de et soit l’ensemble des extensions de degré sur dont la clôture normale est une -extension. Pour chaque discriminant fixé, nous calculons le nombre d’éléments de 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 .
@article{JTNB_2007__19_2_337_0, 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}, number = {2}, year = {2007}, doi = {10.5802/jtnb.590}, zbl = {1161.11034}, mrnumber = {2394890}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.590/} }
TY - JOUR AU - Caputo, Luca TI - A classification of the extensions of degree $p^{2}$ over $\mathbb{Q}_{p}$ whose normal closure is a $p$-extension JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 337 EP - 355 VL - 19 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.590/ DO - 10.5802/jtnb.590 LA - en ID - JTNB_2007__19_2_337_0 ER -
%0 Journal Article %A Caputo, Luca %T A classification of the extensions of degree $p^{2}$ over $\mathbb{Q}_{p}$ whose normal closure is a $p$-extension %J Journal de théorie des nombres de Bordeaux %D 2007 %P 337-355 %V 19 %N 2 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.590/ %R 10.5802/jtnb.590 %G en %F JTNB_2007__19_2_337_0
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, Volume 19 (2007) no. 2, pp. 337-355. doi : 10.5802/jtnb.590. http://archive.numdam.org/articles/10.5802/jtnb.590/
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