Irreducible polynomial and its roots

Question

Suppose I have a polynomial with rational coefficients which is irreducible over the rational numbers.

Let $a$ be one of its roots. Can I express the other roots of this polynomial in terms of $a$ and if so how?

Answer 1

Not in general. Take x^3 - 2 with root a = cubed root of 2. The remaining roots are strictly complex and can not be expressed in terms of a alone (we necessarily need i).