Let M/L/K be a sequence of galois extensions (1-H-G are the corresponding galois groups, so H is normal in G). Also let B>B_l>B_k be primes lying on top of one another. It is unclear to me why the decomposition group of B_l is the image of the decomposition group of B in G/H, now it's clear that this image will be part of the decomposition group but why should it be the whole thing? In other words why is every automorphism of M that fixes B_l the composition of an automorphism fixing H with one fixing B (or the other way around). This seems like maybe a group theoretic argument would suffice but I can't seem to get it. Also a more general question: we know that G acts on the primes lying above B_k, is this action always faithful?