Given $F_q(t)$, we know there are only two non-trivial types of absolute values, which are (1), induced by some irreducible poly (2), induced by the degree valuation.
The completion with respect to the first kind is 'Laurent series' with coefficients in $F_q$. What is the completion with respect to the degree valuation? ($v(f/g)=deg(f)-deg(g)$)
It's just $$\{ \sum_{i=-\infty}^n a_it^i | n\in\mathbb Z, a_i\in\mathbb F_q\}$$ which corresponds to the $\infty$ point on the projective line.