Let $L/K$ be a finite Galois extension of number fields that is ramified exactly at one finite prime and is unramified at all infinite primes. Let $U_K$ and $U_L$ denote the units of the ring of integers of $K$ and $L$, respectively. Some examples show that the norm map on units $Norm_{L^*/K^*}: U_L \rightarrow U_K$ is surjective, or equivalently zeroth Tate cohomology of the group $G$ with coefficients in $U_L$ is $0$. Does this surjectivity of the norm map remain true in general for such extensions?