# Finite extension of the inclusion $\mathbb F_p \to \Omega$ is injective?

Let $$\Omega$$ be an algebraically closed field of characteristic $$p$$ and $$\mathbb F_p \to \Omega$$ be the standard inclusion map. I wonder if for any finite extension $$K/{\mathbb F_p}$$, the extension of this inclusion to $$K \to \Omega$$ is still injective (the existence of the extension is given by the Zorn's lemma,see here for a proof).

To avoid the cyclic reasoning, I hope the solution doesn't use the fact that any two finite fields of the same cardinality are isomorphic.

• Sorry if I'm missing something in your question, but any ring homomorphism between two fields is necessarily injective – Dave Jul 18 at 4:15
• @Dave Okay... what a stupid question.... – No One Jul 18 at 4:17
• Don't worry we all have them now and then, I once asked for a proof of a result and it turned out that intersecting two lines gave a counterexample... – Dave Jul 18 at 4:28