  It is well known that any power series over a finite field represents a
rational function if and only if its sequence of coefficients is ultimately
periodic. The famous Christol's Theorem states that a power series over a
finite field is algebraic if and only if its sequence of coefficients is
$p$-automatic. In this paper, we extend these two results to expansions of
elements in the completion of a global function field under a nontrivial
valuation. As application of our generalization of Christol's theorem, we
answer some questions about $\beta$-expansions of formal Laurent series over
finite fields.
