Field homomorphism and embedding

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

$\varphi(x + y) = \varphi(x) + \varphi(y), \qquad \varphi(x\, y) = \varphi(x)\,\varphi(y), \qquad \varphi(1_F) = 1_G.$

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

• From additivity, $\varphi(0_F) = 0_G$ and $\varphi(-x) = -\varphi(x)$; from multiplicativity, $\varphi(x^{-1}) = \varphi(x)^{-1}$ for $x \ne 0$. These read off because $\varphi(-x)$ is an additive inverse of $\varphi(x)$ and $\varphi(x^{-1})$ a multiplicative inverse of $\varphi(x)$, and inverses are unique (Identities and inverses in a field are unique).
• Every field homomorphism is automatically injective: its kernel $\{x : \varphi(x) = 0\}$ is an ideal of $F$, and a field has only the ideals $\{0\}$ and $F$; since $\varphi(1_F) = 1_G \ne 0$, 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.