Power Residue Criteria for Quadratic Units and the Negative Pell Equation

Let d>1 be a square-free integer. Power residue criteria for the fundamental unit varepsilond of the real quadratic fields mathbb {Q} . 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 mathbb{Q} has an elementary abelian 2-class group and p equiv 5 ... 9 \pmod{16}.