How can I prove these two field extensions are equal?
Q(√3, -√3, i, -i) = Q(√3+i) where Q is the field of the rational numbers.
I got Q(√3+i) ⊆ Q(√3, -√3, i, -i), that direction is easy.
How can I prove Q(√3, -√3, i, -i) ⊆ Q(√3+i)?
Let m ∈ Q(√3, -√3, i, -i)
Therefore m = q + a√3 + bi + c(-√3) + d(-i) where q,a,b,c,d ∈ Q
Let p = a-c and r = b-d. Therefore, p,r ∈ Q
Therefore m = q + p√3 + ri
But I have to prove m = q + p√3 + ri = q + (SOMETHING IN Q)* (√3+i)
Any hints?
I'm trying to prove they're equal by showing each is a subset of the other.
I've shown Q(√3+i) ⊆ Q(√3, -√3, i, -i). Very quickly: m from Q(√3+i) can be written m = q +r(√3+i) = q + r√3 + ri, and let r = a-b,
m= q + a(√3) +a(i) + b(-√3) + b(-i), so m is in Q(√3, -√3, i, -i). Therefore Q(√3+i) ⊆ Q(√3, -√3, i, -i)
I'm trying to prove Q(√3, -√3, i, -i) ⊆ Q(√3+i).
Note: Cleaned up the question a bit.