Affine Galois group of a prime-radical splitting field ($\mathrm{Gal}(\mathbb{Q}({\zeta }_{\ell },{\lambda }^{1/\ell })/\mathbb{Q})\simeq {\mathbb{F}}_{\ell }⋊{\mathbb{F}}_{\ell }^{\times }$)

For distinct or equal prime numbers $\ell ,\lambda$, the polynomial $X^{\ell }-\lambda$ is Eisenstein at $\lambda$. Its splitting field over $\mathbb{Q}({\zeta }_{\ell })$ has Galois group a subgroup of $C_{\ell }$, which is nontrivial because a degree-$\ell$ radical cannot belong to a degree-$(\ell -1)$ cyclotomic field. Thus the full Galois group has order $\ell (\ell -1)$. On roots indexed by $j\in {\mathbb{F}}_{\ell }$, all automorphisms act as $j\mapsto uj+b$. Equality of orders identifies the group with the full affine group over a finite field.

// Target: algebra.bigb