2 problems

Let $\mathfrak{q}$ be a prime ideal with $N(\mathfrak{q})=q\not\equiv1\pmod p$. Suppose that the finite-flat extension group $\operatorname{Ext}^1(\mu_p,\mathbf{Z}/p\mathbf{Z})$ ov…

Let $k$ be a perfect field of characteristic $p$, let $W(k)$ be its ring of Witt vectors, let $K_0=W(k)[1/p]$, and let $K$ be a finite totally ramified extension of $K_0$ with ring…