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 fromto. 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.

Key words: Algebraic function field, finite field, descent of function fields, tensor rank.

1991 MSC: 11G20, 14H25.

Preprint submitted to Elsevier Science : 17 January 2005