My question stems from trying to solve the following problem:
My goal is to calculate the Herbrand quotient of $\mathcal{O}_L^\times$ by hand, so only using elementary calculations in $\mathbb{Q}_p$. If I'm not mistaken it has to be equal to one (which in our lecture we showed for every cyclic extension of local fields).
I already have an idea of how $\mathcal{O}_L^\times$ and $N_G(\mathcal{O}_L^\times)$ look like but still I don't know enough to really calculate the cohomology groups. Can somebody help me out here?