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

by No One   Last Updated July 18, 2019 04:20 AM

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.

Tags :