Galois groups of Kummer extensions of number fields

Let K be a number field, and let G be a finitely generated subgroup of K × . For every prime number ℓ and for all positive integers m,n with m⩾n we show that the structure of the Galois group of the Kummer extension K(ζ ℓ m ,G ℓ n )/K(ζ ℓ m ) only depends on G through parameters that express divisibility properties over K (respectively, over K(ζ 4 ) if ℓ=2, ζ 4 ∉K, m≥2). Moreover, we describe an explicit finite procedure to compute at once the Galois group structure for all extensions K(ζ M ,G N)/K(ζ M ) with M,N positive integers such that N∣M. Our work builds on results in Kummer theory by the last-named author joint with Debry, Hörmann, Perissinotto, Sgobba, and Tronto.