Let $p$ be a rational prime, $K$ be a local field, $K(p) \mid K$ be the maximal $p$-extension of $K$ inside a given separable closure. Now let $I(K(p) \mid K)$ be the inertia group of $K(p) \mid K$ and $U$, resp. $\hat U$ denotes the unit group of the ring of integers of $K$, resp. its pro-$p$-completion.
Local class field theory gives us an isomorphism
$$I(K(p)\mid K)^\text{ab} \cong \hat U$$
where $^\text{ab}$ denots the abelianization.
Can anybody provide me with a good reference for this or explain it to me?
Thank you :-)