Descent of the Definition Field of a Tower of Function Fields and Applications

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-Stichtenoths tower we improve the upper bounds and the upper asymptotic bounds of the bilinear complexity of the multiplication in finite fields.