Some remarks on indefinite binary quadratic forms and quadratic ideals

Abstract

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. In the first section, we give some preliminaries from binary quadratic forms and ideals. In the second section, we obtain some properties of I-gamma and F-gamma for delta = root D, where D not equal 1 is the Extended-Richaud-Degert type (ERD-type), that is, D = w(2)+v for positive integers w and v such that v|4w. In the third section, we obtain some properties of a special family of ideals I-gamma and indefinite quadratic forms F-gamma for gamma = w+1+root D / 2w-v+1