LECTURE 0: JACOBI SYMBOL The Jcobi symbol We wish to generlise the Legendre symbol to ccomodte comosite moduli Definition Let be n odd ositive integer, nd suose tht s, where the i re rime numbers not necessrily distinct Then we define the Jcobi symbol s follows: i ; ii 0 whenever, > ; iii whenever, s Just s in the discussion concerning the Legendre symbol, we begin with some simle roerties of the Jcobi symbol Theorem Suose tht nd re odd ositive integers Then: P i ; ii ; iii whenever P,, one hs ; P iv whenever, P P, one hs ; v whenever P P mod, one hs Proof Prt i is immedite from the definition of the Jcobi symbol, nd rt ii is immedite from the roerties of the Legendre symbol Prts iii nd iv follow directly from rts i nd ii, since the Jcobi symbol tkes vlues 0 or ± For rt v of the theorem, observe tht whenever P P mod, one hs P P mod for ech rime number dividing, whence lso for ech rime dividing The desired conclusion is therefore gin immedite from the definition of the Jcobi symbol
LECTURE 0 Note 3 If the Jcobi symbol, then it follows tht is not qudrtic residue modulo, since for some rime with one must hve tht the Legendre symbol But if, then it is not necessrily the cse tht is qudrtic residue modulo For exmle, one hs, but nd 5 3 5 The Jcobi symbol remins useful for clculting Legendre symbols, becuse it stisfies the sme recirocity nd simlifying reltions s the Legendre symbol s we now demonstrte, nd t the sme time, whenever the Legendre symbol is defined tht is, rovided tht is n odd rime number, then its vlue is the sme s tht of the corresonding Jcobi symbol Theorem 4 Suose tht is n odd ositive integer Then / nd /8 Proof Suose tht is odd, nd tht s with ech i rime number Then i / i i But whenever n nd n re both odd, one hs n n 0 mod, whence n + n n n n n n n mod Iterting the ltter reltion, we deduce tht s i mod, whence / Similrly, we hve i i i i i /8 i But whenever n nd n re both odd, it follows tht whence 8 n n 0 mod, 8 n + 8 n 8 n n 8 n n 8 n n mod
LECTURE 0 3 Thus, gin iterting this reltion, we find tht s i mod, 8 8 i whence /8 Theorem 5 udrtic Recirocity Suose tht P nd re odd ositive integers with P, Then P P /4 P Proof Suose tht q q s nd P r re fctoristions of P nd, resectively, into roducts of rime numbers Then we hve P P r i j q j i j Then by qudrtic recirocity for the Legendre symbol, we obtin P r i q j /4 qj ω, P where we write i j i j ω r i s j i q j i q j 4 But s in the roof of Theorem 4, one hs r s r s i q j i 4 i j q j P mod We therefore deduce tht P P /4, P nd the conclusion of the theorem now follows immeditely Jcobi symbols re useful for clculting Legendre symbols, since they tke the sme vlues for rime moduli, nd one cn ski intermedite fctoristions before lying recirocity Exmle 6 Clculte the Legendre symbol 8093
4 LECTURE 0 One hs 8093 36 0809/4 8093 5 780/4 79 79 4 5 5 So is not qudrtic residue modulo 8093 79 478/4 79 5 Exmle 7 Determine whether or not the congruence x + 6x 50 0 mod 79 hs solution Observe tht x +6x 50 x+3 59, nd hence x +6x 50 0 mod 79 59 hs solution if nd only if But 79 59 0 5 5 79 / 79 79 79 79 79 79 79 4 5 79 /4 5 5 Hence the congruence x + 6x 50 0 mod 79 hs no solution Exmle 8 Let be n odd rime Comute x Let S x There exists such tht For instnce, this is the cse when is rimitive root modulo becuse of Euler s criterion Since,, the m x x mod defines bijection on the set of residues modulo So x x S S Hence, S 0 Counting solutions of congruences For n odd rime nd, b, c Z with,, we consider the congruence y x + bx + c mod Let D b 4c be the discriminnt Theorem The number of solutions with x, y of is equl to: if D, + if D
LECTURE 0 5 Proof We observe tht the number of solutions cn be reresented s the sum We observe tht + x + bx + c + x + bx + c 4x + bx + c x + b D, Since the m x x + b mod defines bijection on the set of residues modulo, we obtin x + bx + c 4 x + b D 4 y D By the roerties of Legendre symbol, We write 4 SD y Then the number of solutions is + When D, SD y 4 y D SD y y This immeditely imlies the second rt of the theorem Suose then tht D We observe tht D SD + z D, z where the sum is crried out over ll non-zero qudrtic residues z We cn lso rewrite this formul s SD z + z D
6 LECTURE 0 Consider the m z z such tht nd z stisfies z z mod Then z z, nd z z D zz D z z D SD + + D z z D + D z z D + + Since the m z D z mod defines bijection, we obtin D z x, nd similrly, z D x Hence, these sums re zero by Exmle 8 This imlies tht SD, which roves the theorem