In the book titled Essays on Numbers and Figures, by Prasolov; it is shown by a simple proof that conjugate root have different form than the original root of the form $m+n\sqrt{3}$.
I feel that the treatment is too primitive, as conjugate is shown of the form $m-n\sqrt{3}$ only.
If the underlying assumption for being a conjugate is that its product with the root gives a rational value; then $-m+n\sqrt{3}$ is also conjugate to $m+n\sqrt{3}$.
Second, the proof that uses equality of $m+n\sqrt{3}$ to $m_1+n_1\sqrt{3}$ is not clear as it is given that $m\ne m_1, n\ne n_1$, & still equality is used for proof.
I feel there should be a better approach shown, that confirms a single negation of signs in the conjugate; as well as to show that $m=m_1, n=n_1$.
In particular, it seems that the answer is given in Galois theory (as given here); but do not know what specific terms the author of the answer is referring to.
As the author is not active on mse for a long time, so request elaboration of the answer.