Is there a method in Galois theory that, say given an $n$th degree polynomial with integer coefficients
$$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$
and $\alpha_1$ is a root of said polynomial, gives another root than can be found using group theoretical techniques? This is clearly true for polynomials irreducible in $\mathbb{R}$ but reducible in $\mathbb{C}$ since many of those come in conjugate pairs but is there a more general method for finding another root using the field operations?