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 Fq2 to Fq. 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.