Field Homomorphism

Definition 1. A field homomorphism is a ring homomorphism between 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 $(S^* \setdif \{0\}, \cdot)$.

Proposition 1. For any field homomorphism $f : F \to K$:

1. \[f(1) = 1.\]
2. \[f(0) = 0.\]

3. For all $a \in F$,

   \[f(-a) = -f(a).\]

4. For all $a \in F \setdif \{0\}$,

   \[f(a^{-1}) = f(a)^{-1}.\]
5. $f$ is always a monomorphism.