If a number defined by radicals is a root of a polynomial with integer coefficients, then so is the same number taking another branch of a root

Question

Suppose x is a number written with an expression composed of rational constants, sums, subtractions, multiplications, division and extraction of n-th roots. That is,

$$x=\alpha^{1/n}\beta+\gamma$$

where $\alpha,\beta,\gamma$ are also of this form, with a finite expression. I want to prove that if $P$ is a polynomial with integer coefficients and $P(x)=0$, and

$$x\prime=\omega^k \alpha^{1/n}\beta+\gamma$$

where k is an integer and $\omega$ is a primitive n-th root of unity, then $P(x\prime)=0$.

I only managed to prove this when $\alpha,\beta,\gamma$ are rational and n is prime. I did it with a sum over the cyclotomic field, which makes me think the general proof will be similar.

I started wondering about this to justify things like the following method to obtain the algebraic conjugates of $\sqrt[3]a+\sqrt[3]b+\sqrt[3]c$:

Answer 1

This is the basic Galois theory. Taking another branch amounts to applying an automorphism of an extension. Every such automorphism takes every root of a polynomial with rational coefficients to another root of that polynomial.