We give an automata-theoretic description of the algebraic closure of the
rational function field F_q(t) over a finite field, generalizing a result of
Christol. The description takes place within the Hahn-Mal'cev-Neumann field of
"generalized power series" over F_q. Our approach includes a characterization
of well-ordered sets of rational numbers whose base p expansions are generated
by a finite automaton, as well as some techniques for computing in the
algebraic closure; these include an adaptation to positive characteristic of
Newton's algorithm for finding local expansions of plane curves. We also
conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version,
includes minor edit
