On the imageℓ-Rank of Abelian Extensions with Restricted Ramification Original Research Article

For a prime number ℓ, we consider ℓ-extensions kS of number fields k unramified outside a finite set Ssubset ofSℓ of places of k, and we study the imageℓ-rank of the abelian part of kS/k and the number of relations of the Galois group GScolon, equalsGal(kS/k).