= Affine Galois group of a prime-radical splitting field
{title2=$\operatorname{Gal}(\mathbb Q(\zeta_\ell,\lambda^{1/\ell})/\mathbb Q)\simeq\mathbb F_\ell\rtimes\mathbb F_\ell^\times$}
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
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