TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A polynomial Galois group acts faithfully on its roots

Statement

Let $0 \neq f \in F [x]$ be separable, let $L$ be its splitting field, and let $X$ be its set of roots in $L$ . The natural action of $G_{f} = Gal (L / F)$ on $X$ is faithful, so it embeds $G_{f}$ in the symmetric group $Sym (X)$ . After ordering $X$ , this gives a subgroup of $S_{n}$ ; changing the ordering conjugates the subgroup.

Facts & Assumptions

Given: The polynomial Galois group of The Galois group of a separable polynomial and the fact that a splitting field is generated over $F$ by its roots (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

[L1]

Every field homomorphism $τ : L \to L$ fixing $F$ maps the finite set of distinct roots of $f$ bijectively to itself (Every $F$ -endomorphism of a splitting field permutes the distinct roots and is an automorphism).

Proof

technique · direct

1.1

By [L1], each element of $G_{f}$ gives a permutation of $X$ , and composition of automorphisms gives composition of permutations.

2.1

If an automorphism induces the identity permutation, it fixes every root of $f$ and fixes $F$ ; because those roots generate $L$ , it fixes all of $L$ . Thus the action homomorphism has trivial kernel and is faithful. For a nonzero constant polynomial, $X$ is empty, $L = F$ , and both groups are trivial; a linear polynomial gives the singleton case.

step 1.1given

3.1

If two orderings of $X$ differ by $π \in S_{n}$ , then the two permutation representatives of every $σ \in G_{f}$ are related by $ρ^{'} (σ) = π ρ (σ) π^{- 1}$ . Hence the embedded subgroup changes only by conjugation.

step 2.1algebra∎

Depends on

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Sources

J. S. Milne, Fields and Galois Theory, v5.10, The Galois group of a polynomial (standard reference, not scraped)
K. Conrad, Galois Groups of Cubics and Quartics, Section 1 (standard reference, not scraped)