Definition of a field homomorphism

Question

Given a field $F$ of characteristic zero, say $F=\mathbb{R}$, what is the minimal requirement for a function $\mu:F\to F$ to be a field homomorphism? (Do we need to require two axioms, one for addition and one for multiplication, or can you encompass everything in one axiom?)

Answer 1

One can insist on two conditions:
(i) $\mu(a+b) = \mu(a)+\mu(b)$
(ii) $\mu(ab)=\mu(a)\mu(b)$
or the single condition
(i)$': \mu(ac+b)=\mu(a)\mu(c)+\mu(b)$,
but the difference is cosmetic. The amount of work involved in checking the combined single condition is the same as in checking the two separate conditions. (Aside: the word axiom is not generally used for conditions defining a homomorphism.)