Note on the quadratic character of a quadratic unit

Abstract

1Φ Introduction* Let m be a positive squarefree integer, and let em denote the fundamental integral unit of the real quadratic field Q(λ/m), so that em = T + £Λ/m with positive integers T and U. Throughout, it is assumed that em has norm — 1, so that m = 1, 5 or 2(mod8), and all odd primes q dividing m satisfy q == I(mod4). A number of recent papers ([1] [3], [7], [9]-[12], [14], [16], [17]) have computed the quadratic character of such em modulo a rational prime p, in terms of representations of a power of p by positivedefinite binary quadratic forms of a certain discriminant associated with m. In this note we prove a result which, among other things, identifies the correct form-discriminant for evaluations of this type. A number of illustrations will be given in § 3 and § 4, after the. proof in §2 of the following theorem.