Let  $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?