Primitive. that ak=1 with K < 4cm ). mod 7. working. then, smaller than 6 will do. m ) =/ odm ) is called. modulo m ( 4) =3. ordz.
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1 R Section 0 Primitive Roots lerned th if ( im cm odm Hover its ossible th k with K < cm For exmle working 7 so if ( 7 L mod 7 know th I ( mod 7 But often n exonent smller thn will do mod 7 : l s 5 ( / I I I Def If mezt nd ( m / KE It such th odm is clled the smllest the order of modulo m denoted ordm ( EI Find the orders of the the bove tble By ordz ( / ( 2 3 ordz ( 3 ordz L mod 7 ordz ( 3 ( orl 7 5 ordz ( o DNE Ord 7 ( 2
2 If use of * Notice th ordm ( E ycm since * But more seems true in the lst exmle (7 the order of ech ( nonzero element divided EI Find ll nezt s t ( mod n 3/ 9/2 ordq 7 7 d wh is ordg (? ( 3 mult order theorem Let me It ( odm if nd only m nd rdf if t divides n cm Becuse know L umiclly get theorem Let me 2? If ( ml ord ( divides cm fotthml for 2 + so tln ( < Assume since 9 nqt odm so n%t T% od m l ( ( mod the division m Assume t with write nq +r lgorithm Now ro show wnt ettre?r#krr@odn n Thus r ( mod m Since is the smllest ositive Since r < t nd rt integer rdm( sit r must not be t I fnodm ositive so s r > io r 0 I
3 EI Find n of the given order if ossible ( ordq ( 2 I ( b ordq ( not ossible ( C ondq ( tril + error / 2 2 EI If ordm ( 2 wh is ordm ( 3? l 2 (3 ordm (d D E _ If cnodm must it be true th ordn (? No : for exmle 0dm but ordm ( 2 Lets revisit Theorem : if C m nd odm < t n>no(modt Thus wh if r r odm # odm ro do odt still set r odt? Recll the ors mod 9 : Note : 5 89 ordg z s 9 mod ( 3 Thus re od 9 2 s r is multile of sr ( mod 3
4 8r Then ( Theorem If ( m nd rdm( re odm if nd only if odt f Since ( ml exists I ( modm We my ssume th szr re s modm < gr Stines times 2 r ties times Stines r ties < + ( s r tines A r s % tem t < s r # odt D 0<7 th EI ( 5 I wh one Suose He ossible vlues for Ords? ( Let <05 k y( 5 Now 5 K must ( 9 5 ( y( Thus be divisor of 2 : ( 2?? n Question : is there with ordy Defy If is lest residue mod m ( for which th is ordm ( cm sy rootm * roots hve the lrgest ossible order
5 use EI 2 is C 9 EI 2 is not root of z 8 7 S ( root of 7 but zig b/w ll G numb 9 rel Prine 9 2 gem (7 2 3 ll Notice th Theorem 5 If 2 why the lest residues g is 5 root of m ( ( b/w I 7 7 numbers rel rice he cre See bove next re exctly # the numbers b/w lnd m th one section sinmy?stimgg2g3 Pd relively rime Theorem M see book Big Question : do roots lwys exist? EI Show th there one not roots of 8 ( E mod 8 The situion is much belter for rimes > Theorem Every Prime hs y ( roots existence but not how finl
6 but 3 sw th non rimes my hve roots e g 9 but not 8 mybe just Prime ors? EI Show 8 hs root Y ( 8 ( z (3 o need find n element with ord oo ( only chnce is with ( 8 / o Use tril + error : odd 5 7 l 5 7 try ^ z next? Ord 3 yes : q( 5 T ord noe g( 7 / 2 : try first? f theorem Let mezt Then hs m existence not root if nd if m 2/ but only Pe or 2e for how find P n rice Ingredients for the roof Then * Remember in The the modulus is Prine Lemm 2 If f ( x ux t + x with n # od f ( x hs most n roots modulo
7 h r + d l t ide Either # \ f is : fc x+ Then ( l so xt hs Sol deg f ( D 72 : if f ( hs root r (x g ( ( mod x P nd gcxo@o# r or f ( D this hs now Y! fer roots bk rice is thn f ( x use induction E _ Show th 2t od hs solutions Lemm } If d / d ( mod hs exctly d solutions * By Ferm s Thn exctly x \ XP ( mod hs solutions ( nmely 2 ( xd ( x d + x 2d+ P xt i K generl form of ( x tx txtl LHS hs ( xd ^( ( x ~ most most d roots by mod roots Lemm 2 xd hs d roots ( nd h( x hs roots d roots
8 I f ide for The WTS th there one ( P roots ot Consider the Set A { 2 } Every element of A hs n order the order must be divisor of nd Let Y( f # of elements of A th hve ordert Note th [ h tl l so lerned in the lst chter th Thus E µe * The gol is show th becuse ( P ( Since * is true it suffices show th C t f(t for ll t T wnt rove Cse : (E O Then clerly Ct< if (e
9 which Notice Let cse 2 : ( t im show YCt ( t nd ll know right now is k > 0 be n element of order Now every element of order t is solution t ( mod nd by Lemm 3 of which is one there re exctly t solutions th ±i (AYEEIE SO 2 3 re the t solutions the elements of order t re on this list ones re they? So hs order 2 : if Lemm k hs order t iff k order t Thus 0 C E # elem of (t D
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