Frobenius Endomorphism on Field is Injective

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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.

$\blacksquare$

Also see

• Definition:Perfect Field