Questions on $\overline{\mathbb{F}_p}|\mathbb{F}_p$ I have some questions on the field extension $\overline{\mathbb{F}_p}|\mathbb{F}_p$ for some prime number $p$: Are the any other intermediate fields besides $\mathbb{F}_{p^n}$ for $n \in \mathbb{N}$ or $\overline{\mathbb{F}_p}$? What is the Galois group of the extension $\overline{\mathbb{F}_p}|\mathbb{F}_p$? Do we have that $\overline{\mathbb{F}_p}= \cup_{n=1}^{\infty} \mathbb{F}_{p^n}$?
 A: Yes, the algebraic closure of $\mathbb{F}_p$ is the union (or more carefully, the direct limit) of the finite fields: first, this makes sense because given any $n$ and $m$, both $\mathbb{F}_{p^n}$ and $\mathbb{F}_{p^m}$ are subfields of $\mathbb{F}_{p^{\mathrm{lcm}(n,m)}}$, so you will indeed get a field. And given any polynomial $f(x)$ with coefficients in $\mathbb{F}_p$, $f(x)$ has at least one root in a finite extension of $\mathbb{F}_p$, hence in the union. And, furthermore, we clearly need at least all the finite fields, since they are the splitting fields of $x^{p^m}-x$ over $\mathbb{F}_p$. So the union is an algebraic closure of $\mathbb{F}_p$. 
The Galois group is the inverse limit of the Galois groups of the finite subextensions (this is true in any infinite Galois extension), which are cyclic; you end up with the inverse limit of the cyclic groups, and that is called $\widehat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$. 
Yes, there are other intermediate fields. For example, take the union of $\mathbb{F}_{p^{2^k}}$ for $k=1,2,\ldots$. These fields form a chain (since $\mathbb{F}_{p^m}\subseteq\mathbb{F}_{p^n}$ if and only if $m|n$), so their union is a field. It is clearly not finite. And it does not contain any element of degree not a power of $2$ over $\mathbb{F}_p$ (in particular, no root of an irreducible cubic), so it cannot be the whole algebraic closure. 
