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

Viviani, Filippo

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

Nous étudions les propriétés de ramification des extensions $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ sous l’hypothèse que $m$ est impair et si $p\mid m$, ou bien $p\nmid v_p\left(a\right)$ ou bien $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(m\right)$ et $v_p\left(a\right)$ 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 $\mathbb{Q}({\zeta }_m,\sqrt[m]{a})/\mathbb{Q}$ under the hypothesis that $m$ is odd and if $p\mid m$ than either $p\nmid v_p\left(a\right)$ or $p^{v_p\left(m\right)}\mid v_p\left(a\right)$ ($v_p\left(a\right)$ and $v_p\left(m\right)$ 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 $m=p^r$.

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