# The Galois orbit of an algebraic number

Let $\alpha$ be an algebraic number and let $S$ be the orbit of $\alpha$ under the action of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$.

Do we have that $\# S$ is bounded from above by the degree of the splitting field of $\alpha$?

-

Yes. The orbit of $\alpha$ consists of the conjugates of $\alpha$, whose number is the degree of $\alpha$, which is a lower bound for the degree of the splitting field of (the minimal polynomial for) $\alpha$.