Quadratic Irrationals, Quadratic Ideals and Indefinite Quadratic Forms II

Authors: Ahmet Tekcan, Arzu Özkoç

Abstract:

Let D = 1 be a positive non-square integer and let δ = √D or 1+√D 2 be a real quadratic irrational with trace t =δ + δ and norm n = δδ. Let γ = P+δ Q 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 = t 2−4n. 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.

Keywords: Quadratic irrationals, quadratic ideals, indefinite quadratic forms, extended modular group.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1330137

References:

[1] R.A. Mollin. Quadratics. CRS Press, Boca Raton, New York, London, Tokyo, 1996.
[2] R.A. Mollin and K. Cheng. Palindromy and Ambiguous Ideals Revisited. Journal of Number Theory 74(1999), 98-110.
[3] R.A. Mollin. Jocabi symbols, ambiguous ideals, and continued fractions. Acta Arith. 85(1998), 331-349.
[4] R.A. Mollin. A survey of Jocabi symbols, ideals, and continued fractions. Far East J. Math. Sci. 6(1998), 355-368.
[5] A. Tekcan and O. Bizim. The Connection Between Quadratic Forms and the Extended Modular Group. Mathematica Bohemica 128(3)(2003), 225-236.
[6] A. Tekcan and H. O¨ zden. On the Quadratic Irrationals, Quadratic Ideals and Indefinite Quadratic Forms. Irish Math. Soc. Bull. 58(2006), 69-79.
[7] A. Tekcan. Some Remarks on Indefinite Binary Quadratic Forms and Quadratic Ideals. Hacettepe J. of Maths. and Sta. 36(1)(2007), 27-36.