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Trying to show a set is a field

  1. Jan 16, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm doing a question where part of it is to show that the span of the following set over the rationals is a field.
    If w is the primitive cube root of unity and z is the positive cube root of 2, the set in question is:

    (1,z,z2,w, wz, wz2)


    2. Relevant equations



    3. The attempt at a solution
    I have already shown this span is closed under multiplication. Almost by definition it is closed under addition/subtraction. I have also shown that each basis element has an inverse in the span. However, when I come to then showing that every element has an inverse, I struggle.. a general element is obviously a linear combination of 6 elements and I can't exactly rationalise the denominator. Any hints at all?

    Thanks,
    Zoe
     
  2. jcsd
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