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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 $\alpha$?

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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) $\alpha$.

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