I am trying to calculate the Galois group of the polynomial f=X4−2X2+2. f is Eisenstein with p=2, so irreducible over Q. I calculated the zeros to be
α1=√1+i,α2=√1−i,α3=−α1 and α4=−α2. Let Ωf=Q(α1,α2,α3,α4)=Q(α1,α2) be a splitting field of f over Q. Since α1α2=√1+i√1−i=√2, we have Ωf=Q(√1+i,√2).
So if we can prove that [Ωf:Q]=8, then we have #Gal(f)=8 and for Gal(f)⊂S4, we must have that it is isomorphic to the dihedral group D4.
How do I go about proving [Q(√1+i,√2)]=8?
Answer
You have shown that your splitting field is K=Q(√1+i,√1−i). The two fields Q(√1±i) are obviously quadratic extensions of Q(i), and these are equal iff (1+i)(1−i)=2 is a square in Q(i), iff √2∈Q(i): impossible. Hence K is a biquadratic extension of Q(i), and [K:Q]=8.
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