Locally finite infinite field

Question

Is there a place (a book maybe) where I can find some useful information on infinite locally finite fields? Especially when all of whose proper subfields are finite?

I know, for instance, that a locally finite field can be embedded in the algebraic closure of a finite field (actually there is an if and only if). However, how can we, for instance find a locally finite field (infinite) whose all proper subfields are finite?

Answer 1

Take a finite field $\mathbb F$, I don’t care what its characteristic is. The Galois group of an algebraic closure over $\mathbb F$ is isomorphic to $\widehat{\mathbb Z}$, the profinite completion of $\mathbb Z$. This group is isomorphic to $\prod_p\mathbb Z_p$, and for any given $p$, it has a quotient isomorphic to $\mathbb Z_p$, thus $\mathbb F$ has an extension $\mathbb K_p$ whose Galois group is isomorphic to $\mathbb Z_p$. This $\mathbb K_p$ is simply the maximal $p$-extension of $\mathbb F$, and it has the property that you seek, that it’s infinite, but every field properly between it and $\mathbb F$ is a finite field.

Answer 2

Explicitly, these fields are all of the form

$$ \bigcup_{n=0}^{\infty} \mathrm{GF}(p^{q^n}) $$

for fixed primes $p$ and $q$. (they can be the same)