# Find the minimal polynomial with real root

1. Mar 26, 2009

### 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

2. Mar 27, 2009

### Hurkyl

Staff Emeritus
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: Mar 27, 2009
3. Mar 27, 2009

### Dick

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.