# A classification of the extensions of degree ${p}^{2}$ over ${\mathbb{Q}}_{p}$ whose normal closure is a $p$-extension

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

Journal de Théorie des Nombres de Bordeaux (2007)

- Volume: 19, Issue: 2, page 337-355
- ISSN: 1246-7405

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topCaputo, 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 19.2 (2007): 337-355. <http://eudml.org/doc/249963>.

@article{Caputo2007,

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 $\mathcal\{E\}_\{k\}$.},

affiliation = {Università di Pisa & Université de Bordeaux 1 Largo Bruno Pontecorvo, 56127 Pisa, Italy, 351, cours de la Libération 33405 Talence cedex, France},

author = {Caputo, Luca},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {-adic field; -extension; Galois group},

language = {eng},

number = {2},

pages = {337-355},

publisher = {Université Bordeaux 1},

title = {A classification of the extensions of degree $p^\{2\}$ over $\mathbb\{Q\}_\{p\}$ whose normal closure is a $p$-extension},

url = {http://eudml.org/doc/249963},

volume = {19},

year = {2007},

}

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

PB - Université Bordeaux 1

VL - 19

IS - 2

SP - 337

EP - 355

AB - 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 $\mathcal{E}_{k}$.

LA - eng

KW - -adic field; -extension; Galois group

UR - http://eudml.org/doc/249963

ER -

## References

top- C. R. Leedham-Green and S. McKay, The structure of groups of prime power order. London Mathematical Society Monographs, New Series 27, 2002 Zbl1008.20001MR1918951
- 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. Zbl0245.12014MR364194
- 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. MR20546
- J. P. Serre, Local fields. GTM 7, Springer-Verlag, 1979. Zbl0423.12016MR554237
- H. Zassenhaus, The theory of groups. Chelsea, 1958. Zbl0041.00704MR91275

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