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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.