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, Volume 19 (2007) no. 2, pp. 337-355.

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 .

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 .

Received:
Published online:
DOI: 10.5802/jtnb.590
Luca Caputo 1

1 Università di Pisa & Université de Bordeaux 1 Largo Bruno Pontecorvo, 56127 Pisa, Italy, 351, cours de la Libération 33405 Talence cedex, France
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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, Volume 19 (2007) no. 2, pp. 337-355. doi : 10.5802/jtnb.590. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.590/

[1] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order. London Mathematical Society Monographs, New Series 27, 2002 | MR: 1918951 | Zbl: 1008.20001

[2] E. Maus, On the jumps in the series of ramifications groups,. Colloque de Théorie des Nombres (Bordeaux, 1969), Bull. Soc. Math. France, Mem. No. 25 (1971), 127–133. | Numdam | MR: 364194 | Zbl: 0245.12014

[3] I. R. Šafarevič, On p-extensions. Mat. Sb. 20 (62) (1947), 351–363 (Russian); English translation, Amer. Math. Soc. Transl. Ser. 2 4 (1956), 59–72. | MR: 20546

[4] J. P. Serre, Local fields. GTM 7, Springer-Verlag, 1979. | MR: 554237 | Zbl: 0423.12016

[5] H. Zassenhaus, The theory of groups. Chelsea, 1958. | MR: 91275 | Zbl: 0041.00704

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