Let $q$ be a prime power, $\mathbb{F}_q$ the finite field of order $q$ and let $\mathbb{F}_q(T)$ be the field of rational functions over $\mathbb{F}_q$.
Similarly to $\mathbb{Q}$, the absolute values on $\mathbb{F}_q(T)$ are indexed by the monic irreducible polynomials over $\mathbb{F}_q[T]$, as well as one at infinity (the "usual" absolute value). So given an irreducible polynomial $F \in \mathbb{F}_q[T]$, let $\mathbb{F}_q(T)_F$ denote the completion of $\mathbb{F}_q(T)$ at $F$.
I have 2 questions:
- Is there a common name for $\mathbb{F}_q(T)_F$? Something like the "field of $F$-adic polynomials"?
- I can see that the way things work in $\mathbb{F}_q(T)_F$ is going to be very similar to $\mathbb{Q}_p$, but have any details about some of the properties of these fields been written out anywhere? I really cant find any reference that uses these fields at all.
It's just called the $F$-adic completion. Your analogy with $p$-adic numbers and $p$-adic integers is slightly off, because "polynomials" would be analogous to "integers" but $\mathbf F_q(T)_F$ is a field, so it's not analogous to $\mathbf Z_p$.
Any account of algebraic number theory that develops it for both number fields and function fields over finite fields via local methods will use the completions of $\mathbf F_q(T)$ and all the other function fields over a finite field. For example, Tate's thesis works for all global fields, and local class field theory does too (but abelian extensions of $p$-power degree for function fields of characteristic $p$ can be tricky). Maybe take a look at Koch's book Number Theory: Algebraic Numbers and Functions.
Most of the basic properties of the fields $\mathbf F_q(T)_F$ are just like those of $\mathbf Q_p$, with nearly the same proofs or literally the same proofs, so you're unlikely to find books going through careful details about the completions of the rational function field $\mathbf F_q(T)$. It's all just an exercise for the reader. :)