With and as in your post take with . Notice that .
Let a be exp(2pi*i/3), b= cubic root of 2 (the real one) and let Q be the field of rational numbers. I want to prove that m:= Q(ab)(b):Q(ab) = 2. I know that m <= 3. And I have proved that m > 1 (that is, b does not belong to Q(ab)). So it is enough to prove that m = 2 or m != 3. However, I cannot find a proof of any of these two statements.
Anybody's hint will be welcome. Thanks for reading.