This note examines the splitting field $K \subseteq \mathbf{C}$ over $\mathbf{Q}$ of the set of polynomials of the form $X^{n}-b$, with $n \in \mathbf{N}^{\ast}$ and $b \in \mathbf{Q}$.We obtain the Galois group $\mathrm{Aut}_{\mathbf{Q}}(K)$ as a natural subgroup of a semidirect product of the Pontryagin dual of a quotient of the divisible hull of the positive rationals and the automorphism group of the maximal abelian extension of the rationals.