Galois cohomology of product

Question

By Hilbert 90 theorem we know that $H^1(Gal(L/K),L^\times)=\{1\}$ for any Galois extension $L/K$. Do we have a formula for $H^1(Gal(L/K),(L^\times)^n)$? Especially when the fields are extensions of $\mathbb{Q}_p$.

Answer 1

Generally speaking, if $A$ and $B$ are $G$-modules then we have a canonical isomorphism $$H^n(G,A\times B)\simeq H^n(G,A)\times H^n(G,B).$$

So if $H^1(G,A)$ is trivial, then $H^1(G,A^n)$ is also trivial.