Find the minimal polynomial with real root

  • Thread starter Daveyboy
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  • #1
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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
 

Answers and Replies

  • #2
Hurkyl
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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:
  • #3
Dick
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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

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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