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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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    $\begingroup$ Sorry if I'm missing something in your question, but any ring homomorphism between two fields is necessarily injective $\endgroup$ – Dave Jul 18 at 4:15
  • $\begingroup$ @Dave Okay... what a stupid question.... $\endgroup$ – No One Jul 18 at 4:17
  • $\begingroup$ 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... $\endgroup$ – Dave Jul 18 at 4:28

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