1. The problem statement, all variables and given/known data Let c be a primitive 3rd root of unity in the complex numbers and b be the real root of x^4-2=0. If a = c*b, show that Q(b,c) = Q(a) 2. Relevant equations 3. The attempt at a solution So [Q(a):Q(c)]=3 and [Q(a):Q(b)]=4, and c and b contain no 'overlapping material', so [Q(a):Q)=12. The usual way I prove things are a simple extension is by starting off by taking (a+b)^2 all the way up to (a+b)^n-1 where n i the deg(a)*deb(b) and then playing around with these expressions trying to come up with a way to get either a or b by itself as to show that a and b are elements of Q(a+b). Since the degree here is 12, things could get quite messy if I use that method. Is there another way to look at this?