Find the minimal polynomial with real root

[Daveyboy]

Find the minimal polynomial with root 2^1/3 + 2^1/2.

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

 

[Hurkyl]

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.

 

[Dick]

Find the minimal polynomial with root 2^1/3 + 2^1/2.

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.