Some Remarks on Indefinite Binary Quadratic Forms and Quadratic Ideals

Let delta denote a real quadratic irrational with trace t = delta + (delta) over bar and norm n = delta(delta) over bar Given a real quadratic irrational gamma epsilon Q(delta), there are rational integers P and Q such that gamma = P+delta/Q with Q|(delta + P) ((delta) over bar + P). Hence for each gamma = P+delta/Q, there is a corresponding ideal I-gamma = [Q, P + delta], and an indefinite binary quadratic form F-gamma(x, y) = Q(x + delta y) (x + (delta) over bary) of discriminant Delta = t(2) - 4n.