Wednesday, 13 November 2013

abstract algebra - Prove that [mathbfQ(sqrt1+i,sqrt2):mathbfQ]=8.



I am trying to calculate the Galois group of the polynomial f=X42X2+2. f is Eisenstein with p=2, so irreducible over Q. I calculated the zeros to be
α1=1+i,α2=1i,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+i1i=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,1i). The two fields Q(1±i) are obviously quadratic extensions of Q(i), and these are equal iff (1+i)(1i)=2 is a square in Q(i), iff 2Q(i): impossible. Hence K is a biquadratic extension of Q(i), and [K:Q]=8.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...