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

Trouble understanding simple Galois Theory example

  1. Feb 25, 2015 #1
    Show that the galois group for (Complex : Reals) is given by {e, y} where y is y: C-->C is the conjugation automorphism defined by y(z) = z~ (Conjugate of z) for all z in C.

    if o is an element of gal(C:R) and z = a + bi in C, then o(z) = o(a+bi) = o(a)+o(b)o(i) = a+bo(i)
    but o(i)^2 = o(i^2) = o(-1) = -1, so o(i) = i or o(i) = -1.

    I am confused on why o(i) can ever equal i... isn't the conjugate of i going to be -1 every time?
     
  2. jcsd
  3. Feb 25, 2015 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Is there a typo at the end of the second paragarph? Your argument that ##o(i)^2 = -1## is fine, but this implies that ##o(i) = i## or ##o(i) = -i## (not ##-1##).

    Then:

    If ##o(i) = i##, then ##o## is the identity, because for an arbitrary element ##a+bi \in \mathbb{C}## we have ##o(a+bi) = o(a) + o(b)o(i) = a + bi##.

    If ##o(i) = -i##, then ##o## is conjugation, because ##o(a+bi) = o(a) + o(b)o(i) = a - bi = \overline{a + bi}##.

    Since these are the only two possibilities, the Galois group consists of the identity and the conjugation automorphism.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Trouble understanding simple Galois Theory example
  1. Galois Theory (Replies: 1)

  2. Galois Theory (Replies: 2)

Loading...