Inflation-restriction exact sequence ($0\to H^1(G/N,M^N)\to H^1(G,M)\to H^1(N,M{)}^{G/N}$)

For a normal subgroup $N$ of $G$ and a $G$-module $M$, the low-degree exact sequence begins $0\to H^1(G/N,M^N)\to H^1(G,M)\to H^1(N,M{)}^{G/N}\to H^2(G/N,M^N)$. The first map inflates cocycles along $G\to G/N$; the second restricts them. If the restriction is trivial, subtract a coboundary to make the cocycle vanish on $N$ and descend to the quotient. Thus for a finite Galois extension, the kernel of restriction on Galois cohomology with finite coefficients is finite.