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.

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