PropositionStatement: 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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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Galois correspondence exchanges composita with subgroup intersections and field intersections with generated subgroups

Statement

Let $K / F$ be finite Galois, and let $E_{i} = K^{H_{i}}$ correspond to subgroups $H_{i} \leq Gal (K / F)$ for $i = 1, 2$ . Then

$Gal (K / E_{1} E_{2}) = H_{1} \cap H_{2},$

and

$Gal (K / E_{1} \cap E_{2}) = ⟨ H_{1}, H_{2} ⟩ .$

Facts & Assumptions

Given: The compositum $E_{1} E_{2}$ , the generated subgroup $⟨ H_{1}, H_{2} ⟩$ of The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, and the subgroup intersection of The intersection of a nonempty family of subgroups of $G$ is a subgroup of $G$ .

[L1]

The assignments $H \mapsto K^{H}$ and $E \mapsto Gal (K / E)$ are mutually inverse inclusion-reversing bijections (The fundamental theorem of finite Galois theory).

Proof

technique · direct

1.1

An automorphism of $K$ fixes $E_{1} E_{2}$ exactly when it fixes every element of both $E_{1}$ and $E_{2}$ , exactly when it belongs to both $H_{1}$ and $H_{2}$ . Therefore $Gal (K / E_{1} E_{2}) = H_{1} \cap H_{2}$ .

L1algebra

2.1

An element of $K$ is fixed by $⟨ H_{1}, H_{2} ⟩$ exactly when it is fixed by every element of both generating subgroups, so $K^{⟨ H_{1}, H_{2} ⟩} = K^{H_{1}} \cap K^{H_{2}} = E_{1} \cap E_{2}$ . Applying [L1] gives the second formula. These membership equivalences also cover equal fields, the base and top fields, and trivial or full subgroups.

L1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

K. Conrad, The Galois Correspondence, Theorem 5.13 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Chapter 3 (standard reference, not scraped)