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Math Help - b algebraic of what degree?

  1. #1
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    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.
    Last edited by ENRIQUESTEFANINI; October 26th 2009 at 08:44 PM. Reason: GRAMMATICAL INCORRECTION
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  2. #2
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    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.
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