Power to Characteristic of Field is Monomorphism

Theorem
Let $F$ be a (finite) field whose zero is $0_F$ and whose unity is $1_F$.

Let the characteristic of $F$ be $p$ where $p \ne 0$.

Let $\phi: F \to F$ be the mapping on $F$ defined as:
 * $\forall x \in F: \phi \left({x}\right) = x^p$

Then $\phi$ is a (field) monomorphism.

Proof
Let $a, b \in F$.

First note that:

So:

Multiplication is more straightforward:

Thus, $\phi$ is a (field) homomorphism.

It is clear that $\phi$ is not a zero homomorphism, since:

Hence, from Ring Homomorphism from Field is Monomorphism or Zero Homomorphism, it follows that $\phi$ must be a monomorphism.

Also see

 * Prime Power of Sum Modulo Prime, where the same technique is used.