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

Conjugate fields and conjugate subgroups of an automorphism group

  1. Jun 1, 2012 #1
    Suppose [itex]E[/itex] and [itex]D[/itex] are both finite extensions of [itex]F[/itex], with [itex]K[/itex] being the Galois closure of [itex]\langle D,E \rangle[/itex] (is this the correct way to say it?) Is it correct that [itex]E[/itex] and [itex]D[/itex] are conjugate fields over [itex]F[/itex] iff [itex]G,H[/itex] are conjugate subgroups, where [itex]G,H\leqslant \text{Aut}(K/F)[/itex] are the subgroups which fix [itex]E,D[/itex]?

    I want to claim also that given [itex]E,D[/itex], we have that their Galois closure [itex]K[/itex] is exactly the field fixed by the core of [itex]\text{Aut}(D/F)[/itex] and [itex]\text{Aut}(E/F)[/itex], but I'm not sure if the "core" is well-defined in this case, since we've not defined a group of which [itex]\text{Aut}(D/F)[/itex] and [itex]\text{Aut}(E/F)[/itex] are a subgroup. What do you think?
  2. jcsd
  3. Jun 4, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    what is the definition of conjugate fields? if you give it precisely maybe you can answer your own question.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook