Vendredi 09 octobre 2020
Marine Rougnant
(Université de Franche-Comté)
What can be said about about the behaviour of a prime along a fields extension of a given field
? After a brief presentation of ramification theory and pro-
groups, we will see that the notion of
-rationality arises naturally when looking at the maximal pro-
extension
of
unramified outside
:
is said to be
-rational when the Galois group
is pro-
free.
Recently, Gras conjectured that a fixed field is
-rational for large
. We will see that the generalized abc-conjecture brings a first step toward the question of quantifying
-rational fields in low degree.