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 $\Frob: F \to F$ is injective.

Proof
We have:
 * $\map \Frob 1 = 1$

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

Also see

 * Definition:Perfect Field