Given that φ ∈ Gal(Q(4root√3)|Q)
Prove that φ(4root√3) = ± 4root√3
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!
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!