= Unramified reduction of a polynomial Galois group
Let $R$ be an integrally closed domain and $f\in R[X]$ monic, with <discriminant> outside a <prime ideal> $P$. The roots stay distinct after reduction. In the splitting algebra choose a prime above $P$. Its <decomposition group> maps onto the <Galois group> of the residue <splitting field>; its <inertia group> fixes every reduced root and is trivial because reduction bijects the distinct root sets. Thus the residue <Galois group> embeds in the generic group as a <permutation> group. For finite residue <fields>, irreducible factor degrees give the <Frobenius cycle type>.
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