For the field $\mathbb{F}_q(t)$, it can be shown that the negation of degree is a valuation, where I defined $\deg 0=-\infty$ and $\deg(f/g)=\deg f - \deg g$.
The corresponding valuation ring $V:=\{r \in \mathbb{F}_q(t) : v(r)\ge0\}$ is the set of all rational functions with degree $\le 0$ and its unique maximal ideal $M$ is the set $\{r \in \mathbb{F}_q(t) : v(r)>0\}$ which is the set of all rational functions with degree $\le -1$.
I am trying to understand the residue field $V/M$. Can it be written as isomorphic to something more explicit or familiar? How do I calculate the quotient?
The quotient field is just $k=\Bbb F_q$. We can think of the valuation ring as the set of $f(t^{-1})/g(t^{-1})$ where $f$ and $g$ are polynomials with $g(0)\ne0$. The valuation ring is defined by $f(0)=0$. The isomorphism $V/M\to k$ is induced by $f(t^{-1})g(t^{-1})\mapsto f(0)/g(0)$.