Definition 1

A field is said to be prime if it is isomorphic either to Q or to one of the fields $\mathbf{F}_p$.

Every subfield of Q contains the ring Z and hence the field of fractions Q of Z; every subring of $\mathbf{F}_p$ is necessarily equal to $\mathbf{F}_p$. Therefore every subfield of a prime field is necessarily equal to it (cf. Cor. 2 of Th. 1 below). Let P be a prime field and A a ring; if f and $f'$ are two homomorphisms of P into A, then the set of $x \in P$ such that $f(x) = f'(x)$ is a subfield of P, hence by what has been said we must have $f = f'$. In particular, the only endomorphism of a prime field is the identity mapping.

$^1$ By abuse of language one says that a ring or field is finite if its underlying set is finite.