What's the minimum number of generators for the wild inertia?

Question

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 $H^1(K^{nr}, \mathbb{Z}/p\mathbb{Z}) \cong H^1(I_w, \mathbb{Z}/p\mathbb{Z})$.

Answer 1

You have the inflation restriction sequence $0\rightarrow H^1(K^{nr}/K,\mathbf{F}_p)\rightarrow H^1(K,\mathbf{F}_p)\rightarrow H^1(K^{nr},\mathbf{F}_p)^{\mathrm{Gal}(K^{nr}/K)}\rightarrow 0$ with the zero on the end because $\hat{\mathbf{Z}}$ has cohomological dimension one. The first term is $\mathbf{F}_p$ by evaluation on Frobenius. The second term can be understood via class field theory. Any continuous homomorphism $G_K\rightarrow\mathbf{F}_p$ factors through the maximal abelian pro-$p$ quotient of $G_K$, which is isomorphic to $\mathbf{Z}_p^{d+1}\times\mu$ where $d$ is the degree over $\mathbf{Q}_p$ and $\mu$ is the group of $p$-power roots of unity in $K$. Since $\mathrm{Hom}_{cts}(\mathbf{Z}_p^{d+1},\mathbf{F}_p)$ is just $d+1$ copies of $\mathrm{Hom}(\mathbf{F}_p,\mathbf{F}_p)$, its dimension is $d+1$. If $\mu$ is trivial, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=0$. If $K$ contains a primitive $p$-th root of unity, then $\mathrm{Hom}(\mu,\mathbf{F}_p)=\mathrm{Hom}(\mu_p,\mathbf{F}_p)$ is $1$-dimensional. So the dimension of the middle piece is $d+1$ or $d+2$ according as $K$ does or doesn't contain a primitive $p$-th root of unity. So the dimension of the last term is either $d$ or $d+1$.