Let $P_\infty$ be the place at infinity at the rational function field $K(x)|K$.
I'm trying to prove that $\deg(P_\infty)=1$ (where $\deg(P_\infty):=[\mathcal{O}_\infty/P_\infty:K]$ and $\mathcal{O}_\infty$ is the valuation ring whose maximal ideal is $P_\infty$).
I know how to find the degree of the place $P_{p(x)}$ associated to an irreducible polynomial $p(x)\in K[x]$ by considering the natural isomorphism $\mathcal{O}_{p(x)}/P_{p(x)}\simeq K[x]/(p(x))$, so that: $$\deg(P_{p(x)})=\deg(p(x))$$ For the case $P_\infty$, I can't see any analogous strategy.