Calcul des groupes de ramification de certaines extensions radicales

For a rational prime $p\mathrm{\ne }2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L\mathrm{/}F$ where $F\mathrm{/}{\mathbb{Q}}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L\mathrm{/}F)$ is a power of $p$, by a Hecke’s theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.