A field homomorphism is a ring homomorphism between two fields.
In other words, a field homomorphism $f : F \to K$ is both a group homomorphism from $(F, +)$ to $(K, +)$ and a group homomorphism from $(R^* \setdif \{0\}, \cdot)$ to $(S^* \setdif \{0\}, \cdot)$.
If $f : F \to K$ is a field homomorphism:
- \[f(1) = 1.\]
- \[f(0) = 0.\]
For all $a \in F$,
\[f(-a) = -f(a).\]For all $a \in F \setdif \{0\}$,
\[f(a^{-1}) = f(a)^{-1}.\]- $f$ is a monomorphism.
Proof.
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(I): By RING-HOM > Definition 1 (III).
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(II), (III), (IV): By GRP-HOM > Proposition 1.
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(V): By (IV) and RING > Proposition 2, $\ker f = \{0\}$, so $f$ is a monomorphism by RING-HOM > Proposition 2.