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Polynomial of degree 4

  1. Feb 11, 2007 #1
    1. The problem statement, all variables and given/known data

    Let R be the field of real numbers and Q the field of rational numbers. In R,[tex]\sqrt 3 [/tex] and [tex]\sqrt 2[/tex] are both algebriac. Exhibit a polynomial of degree 4 staisfied by[tex]\sqrt 2 + \sqrt 3 [/tex].

    2. Relevant equations



    3. The attempt at a solution

    I attempted to construct this by [tex](\sqrt 2 + \sqrt 3)^4 = 20\sqrt 6 + 49[/tex]
    then my polynomial is [tex] x^4-(20\sqrt 6 + 49)=0[/tex]
    is this even close? ThenI have to show it's irreducible and I'm not quite sure where to start that.
    My next problem is to prove that sin 1 is an algebriac number. I'm not sure how to construct that polynomial to prove it's irreducible, either.

    Any hints or advise will be appreciated
    Thanks
    CC
     
  2. jcsd
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