Let $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic 0 .

(i) Let $D$ be a divisor on $X$.

Define $\mathcal{L}(D)$, and show $\operatorname{dim} \mathcal{L}(D) \leqslant \operatorname{deg} D+1$.

(ii) Define the space of rational differentials $\Omega_{k(X) / k}^{1}$.

If $p$ is a point on $X$, and $t$ a local parameter at $p$, show that $\Omega_{k(X) / k}^{1}=k(X) d t$.

Use that equality to give a definition of $v_{p}(\omega) \in \mathbb{Z}$, for $\omega \in \Omega_{k(X) / k}^{1}, p \in X$. [You need not show that your definition is independent of the choice of local parameter.]