Periodic representations in algebraic bases

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We study periodic representations in number systems with an algebraic base $\beta$ (not a rational integer). We show that if $\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(\beta)$ admits an eventually periodic representation with base $\beta$ and digits in $\mathcal A$.