Power to Characteristic of Field is Monomorphism

Theorem
Let $F$ be a (finite) field whose characteristic is $p$ 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:

This means that $\displaystyle \sum_{k \mathop = 1}^{p-1} \binom p k a^k b^{p-k}$ is a multiple of $p$.

So:

Multiplication is more straightforward:

Since $\phi \left({1}\right) = 1^p = 1 \ne 0$, it is clear that $\phi$ is not a zero homomorphism.

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.