Pythagorean, fermatian and P-reducing fields

This paper is devoted to the notion of P-reducing field. which generalizes the notion of a Pytagorean field. We show that if P is an absolutely irredu cible polynomial of Q( T-1, ..., T-n)[X], then there is no proper finite P- reducing extension of the P-reducing closure of an Hilbertian held of chara cteristic 0. In the second part. we study a particular case of P-reducing f ields: ultra-n-Fermatian fields. We show that if p is an odd prime number a nd K is a field of characteristic 0 containing all the p(2)th roots of unit y, then the Galois groups. Gal(K-p(u-term)/K)(where K-p(u-term) is the ultr a-p-Fermation closure of K) is a torsion-free pro-p-group. (C) 2001 Academi c Press.