Can you find a polynomial with integer coefficients...?

Question

Can you find a polynomial with integer coefficients with root(3,3) - root(2,5) as root?

By root(x,n), I mean the nth root of x.

Answer 1

The polynomial you are looking for is of degree 15 and it is not so easy to determine. Here are the steps, I won't do the work.

Let A=root(3,3) and B=root(2,5). Let w be a primitive 3rd root of unity and z be a primitive 5th root of unity. Then w^3=1 and z^5=1. Also, w^2+w=-1, and z^4+z^3+z^2+z=-1.

The roots of the polynomial you are looking for are all of the form
w^i A -z^j B
where i can be 0,1, or 2 and j can be 0,1,2,3, or 4.

Write out (x-r) where r can be each other these roots and multiply all of the terms together. This will give a polynomial of degree 15 which, it will turn out, will have all integer coefficients when you do the correct simplifications.

I can do some of this, by the way. If you multiply out
(A-x)^5 -2,
(wA-x)^5 -2, and
(w^2 A-x)^5 -2,
you will get the correct polynomial. Remember when simplifying that w^3=1, w^2+w=-1, and that A^3=3 and all coefficients will be integers.

Good luck!