  Let us consider an algebraic function field defined over a finite Galois
extension $K$ of a perfect field $k$. We give some conditions allowing the
descent of the definition field of the algebraic function field from $K$ to
$k$. We apply these results to the descent of the definition field of a tower
of function fields.We give explicitly the equations of the intermediate steps
of an Artin-Schreier type extension reduced from $\F_{q^2}$ to $\F_q$. By
applying these results to a completed Garcia-Stichtenoth's tower we improve the
upper bounds and the upper asymptotic bounds of the bilinear complexity of the
multiplication in finite fields.

Number Theory;Algebraic Geometry