A field homomorphism is a ring homomorphism between two fields.
Namely, a field homomorphism $f:F\to K$ is both a group homomorphism from $(R,+)$ to $(S,+)$ and a group homomorphism from $(R^*\setminus\{0\},\cdot)$ to $(S^*\setminus\{0\},\cdot)$.
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 $a\in F\setminus\{0\}$,
\[f(a^{-1}) = f(a)^{-1}.\]- $f$ is a monomorphism.
Proof.
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(I): By RH#DEF (III).
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(II), (III), (IV): By GH#PROP-A.
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(V): By (IV) and RING#PROP-I, $\ker f=\{0\}$, so $f$ is a monomorphism by RH#PROP-MON.