Frobenius Endomorphism on Field is Injective

Theorem
Let $p$ be a prime number.

Let $F$ be a field of characteristic $p$.

Then the Frobenius endomorphism $\operatorname{Frob}:F\to F$ is injective.

Proof
We have: $\operatorname{Frob}(1) = 1$.

By Ring Homomorphism from Field is Monomorphism or Zero Homomorphism, $\operatorname{Frob}$ is injective.