On the Z_l-rank of abelian extensions with restricted ramification

Christian Maire

For a prime number $l$, we consider $l$-extensions $k_S$ of number fields $k$ unramified outside a finite set $S\subset S_l$ of places of $k$ and, we study the $\mathbb Z_l$-rank of the abelian part of $k_S/k$ and the number of relations of the Galois group $G_S:=Gal(k_S/k)$.