NNTDM 7 ( 3-36 A noe on diagonalizaion of inegral quadraic fors odulo Ali H Hakai Dearen of Maheaics King Khalid Universiy POo 94 Abha Posal Code: 643 Saudi Arabia E-ail: aalhakai@kkuedusa Absrac: Le be a osiive ineger be an odd rie and / ( be he ring of inegers odulo Le ( ( n be a nonsingular quadraic for wih ineger coefficiens In his aer we shall rove ha any nonsingular quadraic for ( over ( is equivalen o a diagonal quadraic for (odulo Keywords: Inegral quadraic for Nonsingular quadraic for Diagonalizaion quadraic for odulo rie AMS Classificaion: E8 Inroducion In his secion we sily enion he basic conces of quadraic fors which we shall need hroughou For deails he reader is referred o [] [] and [3] A quadraic for ( over is a olynoial of he ye ( ( a n ij i i j n wih a ij i j n We associae wih ( a syeric n n ari A A given by a a a3 L an a a a3 L a n A a3 a3 a33 L a3n M M M O M an an an3 L a nn Tha is a for i < j ij * * A [ aij ] n n aij a ji for i > j aii for i j Observe ha where ( A j 3
[ K n] M n Here denoes he ransose of he ari On he oher hand noe ha if he ari A is diagonal (An n n ari A is diagonal if a ij whenever i j hen he corresonding quadraic for has he diagonal reresenaion ( A a K annn ie he quadraic for will conain no "cross roduc" ers In he sae way we call a diagonal quadraic for (od for any rie ower if conains no "cross roduc" ers when read (od The deerinan of abbreviaed de is defined o be he deerinan of he ari A We say ha ( is nonsingular over if de Siilarly for any odd rie ower we say ( is nonsingular od if / de Again le ~ be an odd rie ower Le ( and ( be wo quadraic fors over wih associaed arices A A ~ resecively We now view he enries of hese arices as eleens of /( and regard / as he ulilicaive inverse of (od (Alernaively we can relace / wih ( / and regard A as having ineger enries We say ha ( is equivalen o ~ ~ ( (od wrien ( ~ ( (od if here is an inverible n n ari T over /( such ha ~ ( ( T(od ha is A~ T A T (od I is clear ha ~ is an equivalence relaion Noe ha ~ de de (det (od Eale: Le be any odd rie ower and ( where Tha is ( A A ( y aking he sile observaion ha [ ] 443 3 { A Then ( ( (od 443 even 3
we can wrie ( A (od wih A M ( Noe ha since is odd he enries of A are all inegers Thus we ay assue ha A M ( when working wih congruences odulo odd ries Diagonalizaion of quadraic fors odulo Theore For any odd rie ower and nonsingular quadraic for ( over ( is equivalen o a diagonal quadraic for (odulo Proof We roceed by inducion on When i is well known (see [4] ha can be diagonalized over he finie field Say T A (od T D for soe T D M n n ( wih T nonsingular (od and D a diagonal ari Le us lif his o a soluion (od Le U T X where X [ ij ] is a ari of variables We wish o solve This is equivalen o ( T X U A U D D T AT 443 (od A ( T X D (od (od T AT T A X X AT D (od T AT D T A X X AT (od 3 where T A X and ( D T AT / Noe ha is a syeric ari wih ineger enries Le b b b b (od 3 b3 b 33 M M M O bn bn bn3 L b nn Then Thus we are lef wih solving he congruence T A X (od 3
Since T and A are nonsingular (od his equaion has a unique soluion X A ( T (od In he sae anner one can lif a soluion (od o (od for any Indeed roceeding as above suose ha T AT D (od for soe T D M n n ( wih T nonsingular (od and D a diagonal ari Le U T X where X is a ari of variables and solve This is equivalen o U AU D (od ( T X A ( T X D (od T AT T A X X AT D T AT D T A X X A T 443 (od D T AT (od 443 (od where T AX and ( D T AT / is a syeric ari wih ineger enries Le β β β β3 β3 β 33 (od M M M O βn βn βn3 L β nn Then (We noe ha he choice of is no unique Hence we are lef wih solving he congruence T AX (od As T and A are nonsingular (od his equaion has a unique soluion X A ( T (od This colees he inducion se Eales: Le 647 A 48 y ( [ y] y y y Noe ha (y is already a diagonal for when read (od We roceed o diagonalize (y (od 33
34 T I - D T AT Solve (od AX (od (od (od Check : U T X UAU (od Thus ( (od ~ y Le ( y [ ] y y A 48 47 6 (od y Wha haens if A singular? Here A is no inverible so we canno direcly follow he ehod given in our roof Le us ry o solve (od AX T Firs we see ha T I since A is already diagonal (od Le ( a Then
35 A (od and he laer ari has ineger enries A D If we roceed as in he roof we would le (od Now solve X (od This is equivalen o (od (od which give us a conradicion ( and hence here is no soluion of his syse Ne le us ry he choice (od Then (od Solve X (od or equivalenly (od
Le Then obviously (od so ha X Hence i follows one can ake he change of variable a y y a y o diagonalize he quadraic for y ( (od Indeed y ~ y y y ( y y (od (od Our roof of Theore acually yields he sronger resul Corollary If is an odd rie ( is a quadraic for over nonsingular (od n and equivalen o diagonal for a (od hen ( is equivalen o he sae i i i i i n diagonal for a (od for any Noe: This fails for nonsingular fors Indeed ~ (od y bu y ~ / (od i References [] Larry J Gersein asic uadraic Fors Aerican Maheaical Sociey 8 [] G L Wason Inegral uadraic Fors Cabridge Universiy Press 96 [3] Michel Arin Algebra Prenice-Hall New Jersey 99 [4] R Lidl and H Niederreier Encycloedia of Maheaics and is Alicaions Finie Fields Addison-Wesley Publishing Coany 983 36