# Find the minimal polynomial with real root

#### Daveyboy

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

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#### Hurkyl

Staff Emeritus
Gold Member
Try computing $[ \mathbb{Q}(\sqrt[3]{2} + \sqrt{2}) : \mathbb{Q} ]$.

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

Or... something else that would tell you information about the minimal polynomial.

Last edited:

#### Dick

Homework Helper
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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