# Math Help - b algebraic of what degree?

1. ## b algebraic of what degree?

Hi:
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.

2. With $a$ and $b$ as in your post take $f(x) \in \mathbb{Q} (ab)[x]$ with $f(x)= abx^2 + (ab)^2x +1$. Notice that $f(b)=0$.