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[s] is the greatest integer in s; w is an integer in the field QC(a), a= e2 irFr; [1: r] is the integer i in the relation ri -1 (mod p), and if a fraction f/g occurs as an exponent of a, then that exponent is the integer u in the relation fJgu (mod p). In the present paper I shall develop a new line of attack on the Last Theorem by the introduction of power characters in the field Q(e2i1:Ph), h prime to p, in connection with (2). 1. Let n be a prime 0 0 or 1 (mod p) and suppose that xyz 0 (mod n); then (3) xn-1 yn-1 = 0 (mod n). If : is a primitive (n1)th root of unity then in the field Q3() we have (n) = qlq2. . . qlAn-l) where the q's are distinct prime ideals, and So(n-l) is the indicator of n-1. We may take as one of the q's the ideal q = (3 r,n)

Application of the theory of relative cyclic fields to both cases of Fermat’s last theorem