This article is an update of article 170, 1999 Feb 26, with :

• corrections of errors seen by readers or/and myself.
• a new set of congruences of Mirimanoff Polynomials for first case of FLT with study of Fermat equation in \Q(\xi)/K, with \Q \subset K \subset \Q(\xi), see new section (6).
• a new set of explicit polynomials congruences P(t) \equiv 0 mod p for t \equiv -x/y mod p, see new section (7).
• a generalization of results to the diophantine equation x^p+y^p+ c z^p=0, c \in Z, see new section (10).
• for a quick outlook of the content of this version, see Table of Contents, Introduction pp 4-6 and comparative survey with first case bibliography pp 47-52.