Galois groups of Kummer extensions of number fields

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Abstract
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.

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