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Find the minimal polynomial with real root

  1. Mar 26, 2009 #1
    Find the minimal polynomial with root 21/3 + 21/2.

    I would just use maple but I do not have it installed on this machine.
    I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in this case. Do you have another method? I have not tried substituting x=x+1 or x=x-1 or other substitutions.

    The polynomial is x6-6x4-4x3+12x2-24x-4
  2. jcsd
  3. Mar 27, 2009 #2


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    Try computing [itex][ \mathbb{Q}(\sqrt[3]{2} + \sqrt{2}) : \mathbb{Q} ][/itex].

    Or... the dimension of the vector space spanned by powers of [itex]\sqrt[3]{2} + \sqrt{2}[/itex].

    Or... something else that would tell you information about the minimal polynomial.
    Last edited: Mar 27, 2009
  4. Mar 27, 2009 #3


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    You don't need to show the polynomial is irreducible, do you? You just want to show it has no rational roots. Look at the rational root test. If that has a rational root, the root must divide 4.
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