Abstract Linear Algebra?

Given that φ ∈ Gal(Q(4root√3)|Q)

Prove that φ(4root√3) = ± 4root√3

Comments

  • Remember that φ fixes Q.

    If φ(4√3) = a + b * 4√3 for some a, b in Q, then

    φ((4√3)^2) = (a + b * 4√3)^2, since φ respects multiplication

    ==> 48 = (a^2 + 48b^2) + 8ab√3

    ==> a^2 + 48b^2 = 48 and 8ab = 0

    ==> a = 0 and b = ±1.

    So, φ(4√3) = ±4√3.

    (This can be readily checked to yield automorphisms which fix Q.)

    I hope this helps!

Sign In or Register to comment.