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Polynomial of degree 4

  • Thread starter happyg1
  • Start date
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1. Homework Statement

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. Homework 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
 

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