On Galois groups over tamely ramified cyclotomic extensions of algebraic number fields

Abstract

Let 𝑘₀ be an algebraic number field of finite degree, 𝑆₀ be a finite set of primes and 𝐿_{𝑆₀} be the field obtained by adjoining to 𝑘₀ all primitive 𝑞-th roots of unity, where 𝑞 runs over all primes not belonging to 𝑆₀. We shall consider, for an odd prime 𝑙, the maximal unramified pro-𝑙 abelian extension of 𝐿_{𝑆₀} and investigate the structure of this Galois group with certain cyclotomic action.