An inertial degree

Question

Consider the number field $L=\mathbb{Q}[\sqrt[3]{2}]$. If we let $\alpha=\sqrt[3]{2}+1$, then $(3)=(\alpha)^3$ is a prime decomposition. I think that the inertial degree $f((\alpha)|3)=1$ (am I right?). May I ask what is the fastest way to prove this? Thank you.