On the Rationality of the Zeta Function of an Algebraic Variety

Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its prime subfield and is of characteristic p. If T* is the set of all roots of unity in a2 of order prime to p then the restriction of the residue class map to T* is a multiplicative isomorphism of T* onto the multiplicative group of R. The elements of T T* U {O} form the Teichmiiller representatives of A in Q2 and for each x C A the representative of x in Q will be understood to be the element of T in the class x. The non-archimedean valuation of 0 will be denoted by the ordinal function, abbreviated "ord ", and normalized by the condition ord p = 1.