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 $(R^*,\cdot)$ to $(S^*,\cdot)$.
Let $f:F\to K$ be a field homomorphism.
- \[f(1) = 1.\]
- \[f(0) = 0.\]
- \[f(-a) = -f(a)\]
for every $a\in F$.
- \[f(a^{-1}) = f(a)^{-1}\]
for every $a\in F^*$.
- $f$ is a monomorphism.
Proof.
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(I): By DEF-RH (III).
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(II), (III), (IV): By PROP-GH-A.
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(V): By (IV) and PROP-RING-I,
\[\ker f = \{0\}.\]By PROP-RH-MON, $f$ is a monomorphism.