ORBIT OF QUADRATIC IRRATIONALS MODULO P BY THE MODULAR GROUP

Abstract

Let p be an odd prime number, and α be a solution of an irreducible quadratic equation over the rationals Q. In Mushtaq study, the behavior of orbits of a quadratic irrational in a quadratic field 2xaxb0++=()Qα by the special linear transformation group ()SL2,Z modulo 1010,0101⎧⎫−⎛⎞⎛⎞⎨⎬⎜⎟⎜⎟−⎝⎠⎝⎠⎩⎭ is investigated, where; Z denotes the ring of rational integers (Mushtaq, 1988). In this study, the above group is denoted by, presented as the projective special linear transformation group. Let (PSL2,Z α be a root of quadratic equation (mod p), then we shall introduce the orbit of the (irrational) element in a finite field 2xx1−−≡ α[]pFα by, where F equal toZ/.