Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be the wild inertia subgroup (recall that it is pro-$p$). What is the minimum number of topological generators for $I_w$? In other words, what is the $\mathbb{F}_p$-dimension of $H^1(I_w, \mathbb{Z}/p \mathbb{Z})$?

What I would like to compute is $H^1(K^{nr}, \mathbb{Z}/p\mathbb{Z})^{Gal(K^{nr}/K)}$. Does this follow once one knows the answer to the first question? Note that $H^1(K^{nr}, \mathbb{Z}/p\mathbb{Z}) \cong H^1(I_w, \mathbb{Z}/p\mathbb{Z})$.