If $F$ is an algebraic extension of a finite field $K$, then $F/K$ is separable. If we are able to show that $F/K$ is normal, then $F/K$ would be a Galois extension and hence splitting field of a polynomial over $K$, and therefore $F/K$ will be finite. But how to show $F/K$ is normal? Otherwise, if it is not so, give an example such that $K$ is a finite field and $F/K$ is algebraic but $[F:K]$ is infinite.
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Promit Mukherjee
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1A fine question, but handled already. If you want to ask for other (not necessarily closed) infinite algebraic extensions, those can be found inside the algebraic closure. Anyway, I closed this as a duplicate to keep the answers in one place - basically maintaining site hygiene. Welcome to Math.SE! Better luck with the next question. Many "standard" ones have been asked already, so try and search the site also! – Jyrki Lahtonen Apr 14 '20 at 13:10