DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$F$ -homomorphisms and $F$ -embeddings of field extensions

Definition

Let $K / F$ and $L / F$ be field extensions (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions). An $F$ -homomorphism $σ : K \to L$ is a field homomorphism (Field homomorphism and embedding) satisfying $σ (a) = a$ for every $a \in F$ . Because field homomorphisms are injective, it is also called an $F$ -embedding. A bijective $F$ -homomorphism is an $F$ -isomorphism, and an $F$ -isomorphism $K \to K$ is an $F$ -automorphism of $K$ .

Depends on

Used by

Archimedean embeddings and signature Definition
Conjugate algebraic elements over a field Definition
Relative field automorphisms and Aut(K/F) Definition
The separable degree [K:F]ₛ as a count of embeddings into an algebraic closure Definition
Restriction partitions embeddings in a finite tower into extension fibres Lemma
Assuming Choice, a base-field embedding extends across every algebraic extension Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)

FF-homomorphisms and FF-embeddings of field extensions