# Given roots, does there exist a polynomial with "integer" coefficients?

Given roots of form $$a, ~~\sqrt[3]{b+c\sqrt{d}}, \sqrt[3]{b-c\sqrt{d}}$$ $$a,b,c,d\in {\mathbb Z}$$,
does there exist a cubic polynomial with integer coefficients with above roots?

I tried few examples and I seem to find a cubic polynomial every time XD
Is there a proof of this?

NOTE: I came across this while trying to solve this question.

• If $b+c\sqrt d$ is a cube in $\Bbb Q(\sqrt d)$.... Commented Aug 11, 2019 at 17:39
• Have you learned any field theory? This problem designed to be solved with tools of field theory. Commented Aug 11, 2019 at 17:39
• Short answer: yes, as these are all algebraic. Take a polynomial that has the first as a root, multiply it by a polynomial with the second as a root, and multiply by another polynomial with the third as a root. Commented Aug 11, 2019 at 17:39
• @SimplyBeautifulArt How does that ensure integer coefficients... Commented Aug 11, 2019 at 17:40
• @LordSharktheUnknown I'm pretty sure OP is looking for integral polynomials. Commented Aug 11, 2019 at 17:40

Let $$(b,c,d)=(0,1,2)$$. What's the minimal polynomial for $$\sqrt[6]{2}$$?
• I haven't studied abstract algebra, but it seems $x^6-2$ ? I get it! Thank you so much :) Commented Aug 11, 2019 at 17:47