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On some $p$-rational number fields of low degree

par Petit Valentin - publié le

Vendredi 09 octobre 2020
Marine Rougnant
(Université de Franche-Comté)

What can be said about about the behaviour of a prime $p$ along a fields extension of a given field $K$ ? After a brief presentation of ramification theory and pro-$p$ groups, we will see that the notion of $p$-rationality arises naturally when looking at the maximal pro-$p$ extension $K_p$ of $K$ unramified outside $p$ : $K$ is said to be $p$-rational when the Galois group $G_p :=Gal(K_p/K)$ is pro-$p$ free.

Recently, Gras conjectured that a fixed field $K$ is $p$-rational for large $p$. We will see that the generalized abc-conjecture brings a first step toward the question of quantifying $p$-rational fields in low degree.