Field homomorphism and embedding

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

$\phi (x+y)=\phi (x)+\phi (y),\phi (xy)=\phi (x)\phi (y),\phi (1_F)=1_G\mathrm{.}$

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

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