the fontaine-wintenberger theorem relates the absolute galois group of $\mathbb{q}_p(\mu_{p^{\infty}})$ to the absolute galois group of a certain local field over $\mathbb{f}_p$. such an equivalence enables us to understand the galois representation of $\mathbb{q}_p$ with some extra work. in this talk i will explain several key ideas behind its classical proof and its generalization via faltings's almost purity theorem if time permits.