Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Decomposition groups of subextensions

  1. Aug 2, 2010 #1
    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?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted