Field homomorphism

Remarks

So a field homomorphism is a ring homomorphism between two fields. Example: complex conjugation $f(a + bi) = a - bi$ on $\C$ satisfies the three clauses, since conjugation respects sums and products and fixes $1$. Non-example: $f(a) = 2a$ on $\Q$ is additive, but $f(1) = 2 \ne 1$ and $f(1 \cdot 1) = 2 \ne 4 = f(1) f(1)$. Every field homomorphism is injective, by ring homomorphisms from a field are injective. So $f(K)$ is a subfield of $L$ isomorphic to $K$, and $f$ is also called an embedding of $K$ into $L$. Since $f(n \cdot 1) = n \cdot 1$ and $f$ is injective, $K$ and $L$ have the same characteristic. A bijective field homomorphism is a field isomorphism.