ORBIT OF QUADRATIC IRRATIONALS MODULO P BY THE MODULAR GROUP ABSTRACT 2

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1 ORBIT OF QUADRATIC IRRATIONALS MODULO P B THE MODULAR GROUP Shin-Ichi Katayama, Toru Nakahara, Syed Inayat Ali Shah 3, Mohammad Naeem Khalid 3 and Sareer Badshah 3 Tokushima University, Jaan. Saga University, Jaan. 3 Islamia College University, Peshawar (N.W.F.P Pakistan. ABSTRACT Let be an odd rime number, and α be a solution of an irreducible quadratic equation + a+ b over the rationals Q. In Mushtaq study, the behavior of orbits of a quadratic irrational in a quadratic field Q α by the secial linear transformation grou SL, Z modulo, is investigated, where; Z denotes the ring of rational integers (Mushtaq, 988. In this study, the above grou is denoted by PSL(, Z, resented as the rojective secial linear transformation grou. Let α be a root of quadratic equation (mod, then we shall introduce the orbit of the (irrational element α in a finite field F α by PSL, F, where F equal to Z/ Z. [ ] ( INTRODUCTION Let be an odd rime number and be the finite field of elements{,, }. In this case, an element j in the field and the reresentative number j( j in a class { a Z;a j( mod } in the residue class field Z/Z modulo, where Z denotes the ring of rational integers. Q( d be a real quadratic number field over the rationals Q with non-square integer d. In this article, we investigate an analogue in the quadratic etension of the finite field F to a result on the orbits of quadratic irrationals in a global field Q( d(mushtaq, 988. Mushtaq (988 showed Fig. modulo 3, where the diagram is one orbit of length 3 F F in the disjoint orbit decomosition for the quadratic etension F α over the rime F 3 3 field acting on the modular grousl, F. The resent study 3 resents another orbit of length 6 given in theorem. In the figure below, two oints, 8 are fied by, and two oints 4, by in SL(, F 3, where and. To classify the finite field F α according to the number of orbits in the field, where α is a root of a quadratic equation + a+ b ; this study uses Quadratic Recirocity Law to deal with the above mentioned roblem.

2 Katayama et al., Gomal University Journal of Research, -: - ( Fig. Modulo 3 RESULTS AND DISCUSSION Two cases of odd rime numbers were considered, the details of as follows: Case No. :, 4 mod. Let D be the discriminant of the quadratic equation f(. Using the first sulementary and quadratic recirocity law, we have D ±. The equation f is decomosed in the linear factors in F f( ( a( a, + D + c a, where. c a The field F α sα+ t;s,t F coincides with F, namely in the case of, 4 mod, and the field etension F α over F does not occur. ( Let F be the multilicative grou in F, the secial linear transformation grou SL, F, is generated by ( and modulo, in Mushtaq (988. Using the two equations ω ω and ω ω ω ω for ω β ω Q( α, we identify a vector and γ β the ratio for elements β, γ F α. γ β Hence S( β means S for any transformations SL,F. Then ( ω ω, ω By and ω ( ω ω ω 3 ω ω ω. Hence the order of and is and 3 resectively.

3 Katayama et al., Gomal University Journal of Research, -: - (9 As ω ( ω ω ω ω Hence, ( ω ω ω ω+ Then it follows that 3. Therefore, in the case of, 4 mod, we get a single orbit by the action of PSL, F. Case No. :,3 mod. For any rime,3( mod, the discriminant D is not square in F. Thus the field F α sα+ t; s, t F α { } is the quadratic etension over. To determine the orbits by the action of PSL(, F., we roceed as follows: i. For any element a of, and taking the arallel transformation, the closed circuit a a+ a a makes an orbit. ii. Net, assume that a rational element a F and an irrational β F ( α \ F belong to the same orbit. Then there eists a transformation S SL(,F u v such that F F Sa β for β bα+ c,b,c F, we have sa + t β bα+c for β F, ua + v howeverbα + c F, which is a contradiction. iii. Finally, we show that any two irrationals β and γ belong to the same orbit. For two irrationals β bα+ c and γ dα+ f F ( α ; b,c, d,f F, it shows that there S β γ. Taking the arallel transformation : β a β+ denoted by Z. such that eists S SL(,F Since h Z Sb d δ gα for δ gα+h, ut α α. We obtain Sbα dα iff S ( α b dα for S and u v b sb b S SL(,F. ub v Now it is enough to show that sα + t S( α dα with sv tu uα + v for a suitable transformation S, namely ( sα+ t( uα+ v ( uα+ v( uα+ v u ( + uv+ v su + suα+ tu α + tv α su + tu + tv dα gu,v with gu,v u + uv+ v. For d { } d we seek for a rational solution u, v in F such that gu,v d, which imlies that v + uv u + d. ( D u 4 u d u 4d be Let v ( the discriminant of the above quadratic equation on v, then

4 4 Katayama et al., Gomal University Journal of Research, -: - (9 d iii. If is a square e in F, then we find a solution s,t,u,v e,,,e. { } { } iii. We assume that is not square free in for,3 mod, is not square free. Denoting a generator of the multilicative grou F, namely a rimitive root modulo by r. d F By our assumtion, F d is not a square in, assuming the discriminant Dv u + 4d is not a square for any j u r F, we obtained a+ j d+ kj+ r r + r r. kj k + If r + l r, namely k j + k + mod rj r mod j l mod 3 jl. 3 For m m, we have km + d+ r r, namely a+ m d+ d+ a+ r r + r r, hence r r m, which is a contradiction. l, then l, hence (, j l holds for 3 There eists j j such that i kj u r and u + 4d r, we obtain Dv kj r. Finally, we determine the transformation S, with u v u + Dv v, Dv e, where u + e v,e Dv, D u 4d e,e F v + sv tu. If u or v and F, there eists a solution { s, t, u, v} {, u, u, v } { v,,u, v} with or sv tu. In the case, if u v, then + e, hence by e, and by. + 4 d, we get d d, which is a contradiction. Then by the transformation d ( su+ tu+ tv to Z ZS S( α, it was obtained α dα, namely α and d α belongs to the same orbit. Therefore the following theorem was obtained. Theorem. Let be an odd rime and α be a solution of a quadratic equation. Let F α be the field { sα+ t; s, t F } over the finite rime field F {,, }, then: ( For, 4( mod F F we have α and F is occuied by the single orbit of the length by the action of PSL (, Z ;.,3 mod we have the ( For and quadratic etension F α over F F α is searated into two disjoint orbits, namely one is F of the length ;

5 Katayama et al., Gomal University Journal of Research, -: - (9 and the other ( α F \ F of the length by the action of PSL(, F ; the details of these are resented in the diagram below: α α+ α+ α α+ α+ 3 α 3 α+ 3 α+ α α+ α+ α. REFERENCES Kuroki A (7. On quadratic recirocity law. (Bachelor Thesis, Tokushima University, Jaan. Mushtaq Q (988. Modular grou acting on real quadratic fields. Bulletin Australian Mathematical Society. (37: Takagi T (93. A simle roof of the quadratic recirocity law for quadratic residues. Proc. Phys. Math. Soc. Jaan. Ser II, (: Tomonou D (6. Modulser grou which acts on real quadratic fields (Master Thesis. Saga University, Jaan.