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Here's one way: Show that the rational number is an algebraic integer. This may sound like a silly idea, but it has non-trivial applications. A rational number is an integer if it has an expression as a sum of products of algebraic integers. See for example, Prop. 5 in the appendix of Groups and Representations by J.L. Alperin, and Rowen B. Bell, where this method is used as a step in proving that for an irreducible character $\chi$ of a finite group $G$, $\chi(1)$ divides $|G|$.