Maximal unramified extension of a global function field

Question

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?

Answer 1

There aren't any interesting ones. What you're essentially asking for is $\pi_1(\mathbb{P}^1_{\mathbb{F}_q})$. Now, we have the usual short exact sequence

$$1\to \pi_1(\mathbb{P}^1_{\overline{\mathbb{F}_q}})\to \pi_1(\mathbb{P}^1_{\mathbb{F}_q})\to G_{\mathbb{F}_q}\to 1$$

Now, I claim that $\pi_1(\mathbb{P}^1_{\overline{\mathbb{F}_q}})=0$. To see this, note that any (finite) unramified extension $\overline{\mathbb{F}_q}(T)\subseteq K$ corresponds to a finite étale covering $C\to \mathbb{P}^1_{\overline{\mathbb{F}_q}}$. But, the Riemann-Hurwitz formula shows that this map must be degree $1$—explicitly we have the equality $$2g(C)-2=-2 d$$

where $d$ is the degree of the map $C\to\mathbb{P}^1_{\overline{\mathbb{F}_q}}$. But, this clearly implies that $d=1$, so the map is an isomorphism.

Thus, the map

$$\pi_1(\mathbb{P}^1_{\mathbb{F}_q})\to G_{\mathbb{F}_q}$$

is an isomorphism. This tells us that (by picking a section) that all the unramified extensions are of the form $\mathbb{F}_{q^n}(T)$ for some $n$.

EDIT: I didn't notice that you said function fields in general. If your function field is the function field of an elliptic curve then your fundamental group is going to be $G_{\mathbb{F}_q}\times V^{(p)}(E)$ (the prime-to-$p$ Tate module of $E$). For more general curves it's slightly more complicated. You can google the obvious words and lots of things come up.

One suggestion, since you asked this in the number theory section, is to look into geometric class field theory which describes the abelian unramified extensions in terms of quotients of the ideles, like normal class field theory.

Answer 2

The idelic approach to class field theory is not usually considered "explicit". A very approachable paper describing completely explicit generation of abelian extensions of a global function field is in

David R. Hayes, "Explicit class field theory in global function fields"

If I remember correctly, he give you a way to control the conductors, so you should be able to identify the unramified extensions. If you want to read this paper, I suggest starting with Hayes's paper "Explicit class field theory for rational function fields".

Hayes's paper is an alternative to studying the work of Drinfel'd, who obtained the same results in a much more general setting using the machinery of schemes. This work is analogous to the explicit generation of abelian extensions of imaginary quadratic number fields using the theory of complex multiplication.