Abstract
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.
Citation
Stephen Beale. D. K. Harrison. "Generalized cyclotomic fields." Illinois J. Math. 33 (4) 691 - 703, Winter 1989. https://doi.org/10.1215/ijm/1255988579
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