  In this paper we give a general upper bound for the irrationality exponent of
algebraic Laurent series with coefficients in a finite field. Our proof is
based on a method introduced in a different framework by Adamczewski and
Cassaigne. It makes use of automata theory and, in our context, of a classical
theorem due to Christol. We then introduce a new approach which allows us to
strongly improve this general bound in many cases. As an illustration, we give
few examples of algebraic Laurent series for which we are able to compute the
exact value of the irrationality exponent.
