Power Residue Criteria for Quadratic Units and the Negative Pell Equation

Abstract

Abstract. Let d > 1 be a square-free integer. Power residue criteria for the fundamental unit ε d of the real quadratic fields Q( √ d) modulo a prime p (for certain d and p) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation x 2 − dp 2 y 2 = −1. The most important solvability criterion deals with all d for which Q( √ −d) has an elementary abelian 2-class group and p ≡ 5 (mod 8) or p ≡ 9 (mod 16).