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I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following question and couldn't come up with an answer.

Suppose $F$ was a finite field, i.e. $|F| < \infty$. Is it possible to construct an algebraic extension $F'$ of $F$ sucht that $|F'| = \infty$ and $F' \subsetneq \overline{F}$? Meaning, that $F'$ is not algebraically closed.

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  • $\begingroup$ Do you mean that the infinite algebraic extension is properly included in $\overline{F}$ and not $F$ itself? $\endgroup$ – Ulrik Dec 12 '14 at 19:55
  • $\begingroup$ Edited my question. I hope that clarifies the situation! $\endgroup$ – namsap Dec 12 '14 at 20:01
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Fix an algebraic closure $\overline{\mathbb{F}}_3$ and take $F = \varinjlim_{n = 3^k} \mathbb{F}_{3^n}\subseteq\overline{\mathbb{F}}_3$ (i.e., $F = \bigcup_{n = 3^k}\mathbb{F}_{3^n}$). Now, the polynomial $x^2 + 1$ is irreducible over $\mathbb{F}_3$ (it's quadratic, so just test to see that it has no roots). Let $i\in\mathbb{F}_9 = \mathbb{F}[x]/(x^2+1)$ be an element such that $i^2 + 1 = 0$. I claim that $i\not\in F$. To see this, we note that any element $f\in F$ lies in some finite extension $\mathbb{F}_{3^{3^k}}$, and say that no smaller extension of $\mathbb{F}_3$ contains $f$. Then $f$ satisfies a minimal polynomial with $\mathbb{F}_3$ coefficients of degree $3^k$, and as such, $f\neq i$ (because $i$'s minimal polynomial over $\mathbb{F}_3$ has degree $2$).

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  • $\begingroup$ This example readily generalizes to $\mathbb{F}_q$ for arbitrary $q = p^n$, just make an appropriate choice of $i$. $\endgroup$ – Stahl Dec 12 '14 at 20:23
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Sure. Given any "supernatural number" $\pi=\prod p^{e(p)}$ (i.e. a formal product of primes, with no restriction on how many exponents can be nonzero, and with the value $e(p)=\infty$ available) one can speak of the union (within the algebraic closure $\overline{F}$) of all finite fields of order $p^n$ with $n$ dividing $\pi$. If not all of the exponents are zero, i.e. if $e(p)<\infty$ for some prime $p$, then this union will not contain the finite field of degree $p^{e(p)+1}$ over the base field, so it will be proper in $\overline{F}$. Moreover, since the collection of fields that we are taking the union of is directed, the union will itself be a field.

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