DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-25

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.

Field homomorphism and embedding

Let $F$ and $G$ be fields (Field). A field homomorphism $φ : F \to G$ is a map satisfying, for all $x, y \in F$ ,

$φ (x + y) = φ (x) + φ (y), φ (x y) = φ (x) φ (y), φ (1_{F}) = 1_{G} .$

An embedding (or monomorphism) is an injective field homomorphism. An isomorphism is a bijective field homomorphism.

From additivity, $φ (0_{F}) = 0_{G}$ and $φ (- x) = - φ (x)$ ; from multiplicativity, $φ (x^{- 1}) = φ (x)^{- 1}$ for $x \neq 0$ . These read off because $φ (- x)$ is an additive inverse of $φ (x)$ and $φ (x^{- 1})$ a multiplicative inverse of $φ (x)$ , and inverses are unique (Identities and inverses in a field are unique).
Every field homomorphism is automatically injective: its kernel ${x : φ (x) = 0}$ is an ideal of $F$ , and a field has only the ideals ${0}$ and $F$ ; since $φ (1_{F}) = 1_{G} \neq 0$ , the kernel is ${0}$ . This injectivity is used explicitly in Homomorphisms out of a complete ordered field are order-preserving.
Order-compatible homomorphisms are the subject of Ordered-field isomorphism.

Depends on