Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 24F / b / ii / Solution

The inclusion $L((n-1)p)\subseteq L(np)$ allows one additional coefficient: the coefficient of $t^{-n}$ in the local Laurent expansion at $p$, where $t$ is a uniformizer. The resulting linear mapis injective, so the dimension can increase by at most one.

It increases precisely when some rational function has a pole of exact order $n$ at $p$ and no other poles. By the definition of the Weierstrass gap set $G(p)$, this occurs exactly when $n\notin G(p)$. Hence