Quadratic Irrationals, Quadratic Ideals and Indefinite Quadratic Forms II

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 − 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. I. PRELIMINARIES. A real quadratic form (or just a form) F is a polynomial in two variables x, y of the type F = F (x, y) = ax + bxy + cy (1) with real coefficients a, b, c. We denote F briefly by F = (a, b, c). The discriminant of F is defined by the formula b2− 4ac and is denoted by Δ. Moreover F is an integral form if and only if a, b, c ∈ Z and F is indefinite if and only if Δ > 0. Let Γ be the modular group PSL(2,Z), i.e. the set of the transformations z → rz + s tz + u , r, s, t, u ∈ Z, ru− st = 1. Then Γ is generated by the transformations T (z) = −1 z and V (z) = z + 1. Let U = T.V . Then U(z) = −1 z+1 . Then Γ has a representation Γ = 〈 T,U : T 2 = U = I 〉 . So Γ = { g = ( r s t u ) : r, s, t, u ∈ Z , ru− st = 1 } . (2) We denote the symmetry with respect to the imaginary axis with R, that is R(z) = −z. Then the group Γ = Γ ∪ RΓ is generated by the transformations R, T, U and has a representation Γ = 〈 R, T, U : R = T 2 = U = I 〉 , and is called the extended modular group. So Γ = { g = ( r s t u ) : r, s, t, u ∈ Z, ru− st = ±1 } . (3) There is a strong connection between the extended modular group and binary quadratic forms (see [5]). Most properties of Ahmet Tekcan and Arzu Özkoç are with the Uludag University, Department of Mathematics, Faculty of Science, Bursa-TURKIYE. emails: [email protected], [email protected] http://matematik.uludag.edu.tr/AhmetTekcan.htm. binary quadratic forms can be given by the aid of the extended modular group. Gauss (1777-1855) defined the group action of Γ on the set of forms as follows: Let F = (a, b, c) be a quadratic form and let g = ( r s t u ) ∈ Γ. Then the form gF is defined by gF (x, y) = ( ar + brs+ cs ) x +(2art+ bru+ bts+ 2csu)xy (4) + ( at + btu+ cu ) y, that is, gF is gotten from F by making the substitution x → rx+ tu, y → sx+ uy. Moreover, Δ(F ) = Δ(gF ) for all g ∈ Γ, that is, the action of Γ on forms leaves the discriminant invariant. If F is indefinite or integral, then so is gF for all g ∈ Γ. Let F and G be two forms. If there exists a g ∈ Γ such that gF = G, then F and G are called equivalent. If detg = 1, then F and G are called properly equivalent and if detg = −1, then F and G are called improperly equivalent. A quadratic form F is said to be ambiguous if it is improperly equivalent to itself. An indefinite quadratic form F of discriminant Δ is said to be reduced if ∣∣∣√Δ− 2|a|∣∣∣ < b < Δ. (5) Mollin (see [1]) considered the arithmetic of ideals in his book. Let D = 1 be a square free integer and let Δ = 4D r2 , where r = { 2 D ≡ 1(mod 4) 1 otherwise. (6) If we set K = Q( √ D), then K is called a quadratic number field of discriminant Δ = 4D r2 . A complex number is an algebraic integer if it is the root of a monic polynomial with coefficients in Z. The set of all algebraic integers in the complex field C is a ring which we denote by A. Therefore A ∩K = OΔ is the ring of integers of the quadratic field K of discriminant Δ. Set wΔ = r − 1 +√D r for r defined in (6). Then wΔ is called principal surd. We restate the ring of integers of K as OΔ = [1, wΔ] = Z[wΔ]. In this case {1, wΔ} is called an integral basis for K. Let I = [α, β] denote the Z-module αZ ⊕ βZ, i.e., the additive World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:11, 2009 1063 International Scholarly and Scientific Research & Innovation 3(11) 2009 scholar.waset.org/1999.7/2294 In te rn at io na l S ci en ce I nd ex , M at he m at ic al a nd C om pu ta tio na l S ci en ce s V ol :3 , N o: 11 , 2 00 9 w as et .o rg /P ub lic at io n/ 22 94 abelian group, with basis elements α and β consisting of {αx+ βy : x, y ∈ Z}. Note that OΔ = [ 1, 1+ √ D r ] . In this case wΔ = r−1+ √ D r is called the principal surd. Every principal surd wΔ ∈ OΔ can be uniquely expressed as wΔ = xα+ yβ, where x, y ∈ Z and α, β ∈ OΔ. We call α, β an integral basis for K, and denote it by [α, β]. If αβ−βα √ Δ > 0, then α and β are called ordered basis elements. Recall that two basis of an ideal are ordered if and only if they are equivalent under an element of Γ. If I has ordered basis elements, then we say that I is simply ordered. If I is ordered, then F (x, y) = N(αx+ βy) N(I) is a quadratic form of discriminant Δ (Here N(x), denote the norm of x). In this case we say that F belongs to I and write I → F . Conversely let us assume that G(x, y) = Ax +Bxy + Cy = d(ax + bxy + cy) be a quadratic form, where d = ±gcd(A,B,C) and b2−4ac = Δ. If B2−4AC > 0, then we get d > 0 and if B2−4AC < 0, then we choose d such that a > 0. Set