Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$

Viviani, Filippo

doi:10.5802/jtnb.470

Ramification groups and Artin conductors of radical extensions of

ℚ

Viviani, Filippo ¹

¹ Universita’ degli studi di Roma Tor Vergata Dipartimento dimatematica via della ricerca scientifica 1 00133 Roma, Italy

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 779-816.

Résumé
Abstract

Nous étudions les propriétés de ramification des extensions $ℚ (ζ_{m}, \sqrt[m]{a}) / ℚ$ sous l’hypothèse que $m$ est impair et si $p ∣ m$ , ou bien $p ∤ v_{p} (a)$ ou bien $p^{v_{p} (m)} ∣ v_{p} (a)$ ( $v_{p} (m)$ et $v_{p} (a)$ sont les exposants avec lesquels $p$ divise $a$ et $m$ ). En particulier, nous déterminons les groupes de ramification supérieurs des extensions complétées et les conducteurs d’Artin des caractères de leur groupe de Galois. A titre d’application, nous donnons des formules pour la valuation $p$ -adique du discriminant des extensions globales considérées avec $m = p^{r}$ .

We study the ramification properties of the extensions $ℚ (ζ_{m}, \sqrt[m]{a}) / ℚ$ under the hypothesis that $m$ is odd and if $p ∣ m$ than either $p ∤ v_{p} (a)$ or $p^{v_{p} (m)} ∣ v_{p} (a)$ ( $v_{p} (a)$ and $v_{p} (m)$ are the exponents with which $p$ divides $a$ and $m$ ). In particular we determine the higher ramification groups of the completed extensions and the Artin conductors of the characters of their Galois group. As an application, we give formulas for the $p$ -adique valuation of the discriminant of the studied global extensions with $m = p^{r}$ .

MR Zbl

DOI : 10.5802/jtnb.470

Affiliations des auteurs :

Viviani, Filippo ¹

¹ Universita’ degli studi di Roma Tor Vergata Dipartimento dimatematica via della ricerca scientifica 1 00133 Roma, Italy

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     title = {Ramification groups and {Artin} conductors of radical extensions of $\mathbb{Q}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {779--816},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
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}

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Viviani, Filippo. Ramification groups and Artin conductors of radical extensions of $\mathbb{Q}$. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 779-816. doi : 10.5802/jtnb.470. http://www.numdam.org/articles/10.5802/jtnb.470/

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