Christol's theorem
Automatic sequences and formal algebraic series

In this section, we will show Christol's theorem, which states the equivalence between the algebraicity of a formal series with coefficients in a finite field and the automaticity of the sequence of its coefficients: in other words, on a finite field, a combinatorial property of the sequence of coefficients of a formal series (the fact of being generated by a finite automaton) allows us to detect the algebraic character of this formal series.