Quadratic irrationals, quadratic ideals and indefinite quadratic forms II

Abstract

Let D ≠ 1 be a positive non-square integer and let or be a real quadratic irrational with trace t = δ + δ and norm n = δδ. Let be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P] and an indefinite quadratic form Fγ(x, y) = Q(x-γy)(x-γy) of discriminant Δ = t2 -4n. In the first section, we give some preliminaries form binary quadratic forms, quadratic irrationals and quadratic ideals. In the second section, we obtain some results on γ, Iγ and Fγ for some specific values of Q and P.