^ j> T - -i- ~<_^ be any two polynomials and l e t Consider f i r s t the case i n which ^ \u2014 \/ \u2022 Divide J-c\u00a3> by as i n the ordinary d i v i s i o n process u n t i l we get a remainder of lower degree than ^P^j) and having the quotient <^^) \u2022 Then } \/(j) = gCj) - ) from which (1) follows. Consider the case where ^ \/ and ^= O , mod ~7T~. Let ?^ be the reciprocal, mod 7T~ , of ^ a Then jr \/V S j \u2014 where flfg) is a polynomial i n ^ having unity as the coeffi c i e n t of the term of highest degree, Divide by ^ j ^ J as before. Then f**\u00a3<Cj)+ - ^ ^ ^ where 9^3, (j) and a r e t h e quotient and remainder required. XI. Congruence of Two Polynomials With Respect to a Double Modulus. Two polynomials f,(j) \u2022\u00bb \/ ^ ^ J a r e s a i d *\u00b0 ke i d e n t i c a l l y congruent to each other with respect to the double modulus 7T, \u00bb where T is a prime of and ft\/j) a polynomial, i f their difference, fj)'- f*cj) i s d i v i s i b l e by \u00bb m O Q \u2022 where <=2(j) and are polynomials. If J - ) i s d i v i s i b l e , mod ^T\" by this may be 16. expressed by ffy) = O , -^^^ 77*^ > \u2022 This i s an immediate consequence of the de f i n i t i o n of d i v i s i b i l i t y * Theorem 5* If J~' (\u00a3) > f-*. ) b e a n v * w o poly-nomials and \"77~a prime of \u2022$(<) , there exists a common divisor, , mod , of J~< (j) ? (j) such that j ) ^ ) is d i v i s i b l e , mod 7T-, by every common divisor, mod ~T~ \u201e of f' ^ j) 3 J-*- CJ^ * a n <^ \"there exist two polynomials Cj) such that Assume that -f-^ ^ ) i s of degree not higher than \/\/ ^ ) \u2022 Then we may obtain two polynomials c\u00a3,(j) and \/s^jJ such that J^3 ^) being of lower degree than j-^ (^) . Dividing y \u00a3 ^ Cj) where * i s of degree lower than fs (j) Similarly and since the degree of each remainder i s decreasing we must f i n a l l y , after a f i n i t e number of steps reach a remainder ' J[M_f\/ (g. ) which i s zero, mod 77~ . Now the common divisors, mod , of r ) and -f^ Cj^) are identical with those of ^ ) and jz^.^ and so on u n t i l f i n a l l y those of \u00a3^ , (j)- and ^ 3 with those of a n d \u00a3 (\u00a7)\u2022\"\u2022 But ^ ; i s a common divisor, mod of J^.^) and a n d i s evidently d i v i s i b l e by every common divisor of f^tj) and \u2022 Hence ) i s the required common divisor, ~J)(j) > mod 17- , of fi^J and A<\/; \u2022 Now substitute for J~s ^ ) * n t n e second con-gruence i t s value i n terms of \u00a3(j) a n d y \u00a3 ) found from the f i r s t congruence, s i m i l a r l y the values of -f3 (j) andJ-\/^) i n the third congruence their values i n terms of f, fj} > fa. (j ~> i s v i s i b l e , mod IT , by r~^)\u00a3?J and suppose that \u00a3 fj) i s not d i v i s i b l e j mod If, by <~l~>^)-Then there exist two polynomials, ^)i Cor 4: If the product of any number of polynomials be d i v i s i b l e , mod ~n~ , by a prime polynomial ) , then at least one of the polynomials is d i v i s i b l e , mod ~77~t byHP^^-Cor 2: If neither of two polynomials be d i v i s i b l e , mod ~n~ by a prime polynomial ) , their product is not d i v i s i b l e , mod 7T \\ by \" P \u00a3 ) . Theorem 7. A polynomial \u00bb can be resolved i n one and but one way into a product of prime polynomials, mod ~TT, Let be a polynomial of degree , mod lf~ and in i t s reduced form, mod 77\"~. Then A^J.) i s either prime or has a divisor, ^^j) say, mod 77~ . I f is prime, the theorem i s evident. If i t is not prime then where the sum of the degrees of^^J^ and is ~H and neither i s a unit. If i s not a prime polynomial i t must have a factor fi^l \u00bb m o d \u00bb s o *hat where neither ^?^)or yr (j) i s a unit and the sum of the degrees of (j) and X, (j) is the same as the degree of If (j) i s not a prime proceed as before. Since the degrees of the factors are decreasing we must, after a f i n i t e number of steps reach a prime polynomial 20. T?^) \u00bb mod W~* Then Proceed i n the same manner with ^ (j ) \u2022 I f i t i s not 'prime we must f i n a l l y obtain where f^\u20ac. i s f i n i t e . To show this factorization i s unique, assume that we have Then -<\u2022* I J~? Then at least one of the eS. (^) , say \u00ab ^ ) , must be d i v i s i b l e , mod , by^Tf (^) and hence, since they are prime, must be associated, mod 7T~ where is a unit, mod \/\/ . 21, Proceeding as before we may show that for each ^ ) there i s associated, mod 7T~, at least one Sj (p . More over A f two or more T^fj)?? are associated with each othe 'at least as many ^) be the quotient. Continuing as before* we see whether i s d i v i s i b l e by any of the remaining prime polynomials of the f i r s t 22 . degree. F i n a l l y we obtain where* y4l ^ ) contains no factors, mod 77^ of degree less \"than the second. To find the prime factors, mod 7T~, of the second degree we determine, i n the same manner as above, which prime polynomials of the second degree divide (j) , mod 77\" , and si m i l a r l y for those of the third degree, etc. If , however, we do not know the prime polynomials of the second degree we may determine whether ) i s d i v i s i b l e , mod 7Tr by any polynomial of the second degree. If i t i s , then such a polynomial must be prime since contains no factors of degree less than the second. The same applies for polynomials of higher degree. X1T. General Congruence 2\u00a3th Degree in One Unknown. Theorem 8, I f b e a root of the congruence ^-(^) i s d i v i s i b l e , mod 7r~, by J-y\u00b0 * a R d conversely i f \/^j) i s d i v i s i b l e , mod 7~, b y ^ - y \u00b0 \u00bb thenyois a root of (1), Dividing y ^ ; by ^ ~ \/ \u00b0 * m o d ~ \u00bb w e obtain If j\u00b0 i s a root of (1) then J~^\/\u00b0J == O , mod w Hence \/^^ [ t t ] roots and hence i f <^ t, - j \u00b0<^ w constitute a complete residue system* mod ~TT~, we have the ident i c a l congruence s ( \/ \" - - c \/ - ^ ' \u2122 ^ o or, since JT is a prime, we may chose as our residue 25. system, mod ~TT , the -n. JV2 rational integers \u00b0) j [ V ] - \/ ? and so XVI* Analogue to Wilson's Theorem i n If 7T be a prime of \/fVVJ and \/\u00b0> P, P , be a reduced residue system, mod 7T\", then We have Let 5^ be the sum of a l l possible products of \/J, \u00bb \/ ^ v w taking j at a time. Then - \/ ^ S,>^L--'--~+C-O^S-w Equate coefficients of l i k e powers of \u00a3 . Then Now since [ V ] - 77~7r 7 i s always odd except when \/T-= \/+u or i t s associates, ^ Ctt) =\u2022 \">c \u00a3-*-\"] -\/ i s even and therefore When = \/?\u00ab\u2022-*'_, or i t s associates, = -2. ^?(tr) = and we see that the theorem i s also s a t i s f i e d . The theorem may also be proved d i r e c t l y by putting ^ = o i n the congruence (1) and proceeding as above* 26. XVII. Common Roots of Two Congruences. The common roots of two congruences .\/> fj ) Therefore, in order to find the incongruent roots of any congruence with prime modulus, we need only find the roots of the congruence where is \"the greatest common divisor, mod ir~% of the given congruence and the congruence This follows since the l a t t e r congruence has for i t s roots the roots of a complete residue system, mod ~TT\\ XVIII. Determination of Multiple Roots of a Congruence with Rrime Modulus. Theorem 11. I f the congruence has a multiple root yc? of order ~g , the congruence has a multiple root of order For l e t be a prime polynomial, mod 7T~ and suppose f^J) is d i v i s i b l e , mod 77~, b y p p ^ V j but not n -> -\u00a3+\/ by f p c ^ ) ] . 27. Then where and 7 ^ ; are polynomials, and T^j) and f-^j) 'are prime to each other, mod 7T . Also where ) ? gt '(^y ^ ^ ) are polynomials i n ^ -Therefore where i s a polynomial. Moreover i s not d i v i s i b l e , mod 1F~} by Tpf^) since ^ '(j) i s of degree less than ~P^) and <2<^ ) i s prime to \"P^J \u00bb mod TT \u2022 Theref ore f(j) i s d i v i s i b l e , mod 7T, by j^~p ) ~j but not by f f P ^ j ] ^ . In p a r t i c u l a r , i f ; has a root, JD , of order , then j-kj) has a root, y\u00b0 , of order t f - \/ \u2022 XIX. Congruences i n One Unknown and with Composite Moduli. To find the solution of a congruence of the form where ^ Case I. When^\/t. are integers of 7eU) which are prime to each other the solution of (1) may be reduced to the 28. solution of a system of congruences This follows since every root of (1) obviously s a t i s f i e s each of the congruences (2) while any integer of which s a t i s f i e s simultaneously the system of congruences (2) must s a t i s f y (1). If then, p. be the roots of the congruences (2), and i f \\\\ be any integer of &<) such that then \u00a3 i s a root of the congruence (1). The system of congruences (3) i s always solvable by the method of \u00a7 VI . I f any one of the congruences (2) has no root, then (1) has no root. Case I I . When^^ =- \/r- .. 77^ the congruence (1) may re be Asolved into the V congruences The solution of such a congruence may be made to depend upon the solution of one of the form where the power of the modulus i s one less than in the ori g i n a l congruence, and so f i n a l l y may be made to depend upon the solution of a congruence of the form where 77^ i s a prime modulus. For l e t ^ be a root of (5). Then a l l integers 29. of the form J= + IT. ^ where 7 i s an integer of &j) , are roots of (5). Since a l l roots of (4) are roots of (5), any\/root of (4) must be of this form. Then ^ \u2014 77 or and hence since -f(^) == -~^<^~&\u00a3 TT~ * '} \/ ( p = K 7 7 -and dividing each term of (6) by 77~ \u2022' ' we find This i s a necessary and s u f f i c i e n t condition that ^ must sa t i s f y i n order that a root of (5) may also be a root of (4). (a) If -f '(f) j\u00a3 O, then there is one and but one value of ^ which w i l l s a t i s f y (7) and so one and but one value J\" \u2022+- y 7T\\ which s a t i s f i e s (4). (b) If f 'CfJ c>j -^rt^Tir: j v<- ^ \/< ^ O-^^TT there is no value of y satisfying (7), and hence no value of ^ of the form j -+- ~7T^. y satisfying (4); that i s , (4) has no root. (c) I f f'ff* \"\u2122<^\"7 j G^ct K=~ o -^^77* then (7) i s an ident i c a l congruence and consequently has ^ \u00a37\/7] solutions, mod 7TT , from which we may find ^C'TT] solutions of (4). 30. Example: Solve the congruence This may be made to depend upon the solution of which has as a root ^ ^ \/ } ~^^.^-^ \/ - ?^-e' -Then the roots of (1) are of the form Substituting i n (1) we obtain This gives \/ as the only root of ( 1 ) , XX. Residue of Powers* I f i s prime t o ^ and i f ^ =\u2022 <=<: , mody* , where \u00a3~ is a positive rational integer, then\/^ is called a power residue of \u00ab?< with respect to the modulus^*'. A system of integers of such that every power residue of * mod^^ , is congruent to one and only one integer of the system, modyfc * i s called a complete system of power residues of <\\ with respect to the modulus >-y\u00a3 \u2022 Consider the congruence ^ =\u2022 \/, \u2014x^-^^c CO It i s evident from Fermat*s theorem that \/\"always exists and that ^ PC**-) * T n e integer ^ i s said to appertain to the exponent when \u2022 i s the smallest value of jf~, other than zero, for which the congruence (1) i s true. 31* I f \u00bb mod^, then <=< and ^3 appertain to the same exponent, mod^ . Theorem 12. If the integer <=>< appertains to the exponent , mod^ , then the --^ powers of ^ , ^ \u00b0 = o*', ^ ; ^ c ^ - ^ _ \/ co are incongruent each to each, modM * Let oA' \u00bb according as their exponents are congruent or incongruent, mod * Let o< ' by any two powers of o< where 5\"(J 5^ are positive integers, and l e t S, = ^ G.+r; \u2022 , ^< , <'> and ^ being positive integers and osr**^ > o * a ^ ^ , -r^ri & Assume that Then and since i s prime to 32* But, since O ^ K-1 < by ( 2 ) , and o< appertains to the exponent , then \u2022n - ri = o Therefore, from (1) i s a necessary condition for (3) to be true. Moreover, i f (8) i s given, then from (1) r, = n- > \u2014 \u2014 ^ and hence since r and y? are both less than \u00a3 we must have Then and since we have Therefore (6) i s also a s u f f i c i e n t condition that (3) be true. Moreover The rel a t i o n (7) expresses the law of the p e r i o d i c i t y of power residues. 33. Theorem 14. The exponent to which an integer <=< appertains, mod^r, i s a divisor of \u00a37^) For\"* and, therefore, by theorem 13, i . e . jZtffa-) i s d i v i s i b l e by -Theorem 15. I f two integers, ^ , and \u00a9< a , appertain, modyt , to two exponents tit^ , , respectively, that are prime to each other, then their product, <= \u00b0 * J O < , - - > ^ which, by theorem 12, are incongruent to each other, mod 7Tt i s a root of the congruence J ^ \/ j (A) For i f cxtf be one of them, then since ^ \/ ^ \u2014 -These are, moreover, a l l the incongruent roots of (2) since (2) cannot have more roots than i t s degree, being prime to ~7T~. Now l e t o< be one of the integers (1) and suppose that b>\u00b0< i . e. where A- 6 may have values from \/ to (^CTTJ - \/ In particular H e n c e i s a primitive root of ~77~. We may therefore define a primitive root of '\/\/ as an 3 7 . integer of fa) such that a complete system or i t s power residues, mod 7T , constitute a reduced residue system, mod* 77\". We sh a l l now give a second proof of Wilson's theorem depending upon this d e f i n i t i o n . Let F f e e a prime of such that ^ QTTJ is odd, and l e t j\u00b0 be a primitive root of i r and f, } > f^rJ a reduced residue system, mod y r . Then since y^\/ 0* \u2014 j\/\u00b0 a l s o constitute a reduced residue system, mod 7T~ T we have and are the integers \/, .2, r) i n some order. Multiplying these congruences together we have Ht* ^ . .'- T T ^ s f> , ~\u2122~^7r-But since we must have 38. Therefore \u2022^ IFTT} is always odd except when ~7T~=} or ' i t s associates, i n which case JJTT] = 3. , The proof as given here does not apply for this case although the theorem i s true since then y?f*-J =\u2022 \/ , and we take \/ as our reduced residue system, mod \/-*\u2022\u2022\u00a3. . XXII. Indices. Let P be a primitive root of 7T~ . Then f, f, - - -\u2014 f\u00b0 constitute a reduced residue system, mod \/\/ , so that any number \u00b0< which is prime to 7T~must be congruent to some power of j\u00b0 , mod TT . If then i s said to be the index of <=>< to the base y\u00b0 , mod 7T~T and we write ^ =\u2022 ^ ^^^^ *- <\u2014^^> so that Let c< be an integer of such that Then and hence 40. In p a r t i c u l a r , putting o< =\u2022 jo , we have i . 6T. . \/? = \/ , \u2014 Theorem 17. If <=<: , mod 77\", be , and is the greatest common divisor of -c' and ^\/Vy 1 , then <=< appertains to the exponent ^Vj\"\/^ We are given that \u00ab-<' ~ j\u00a3> ^ ^-^^<\u00bb-0e- 77\u2014~ Let be the exponent to which <=* appertains, mod 7i~ . We wish to find the least value of J- such that Now and hence > Then where i s the greatest common divisor of -c' and . Then ^ 7 - i s prime to and so the smallest value of j- , other than zero, such that ( 2 ) be s a t i s f i e d i s Then, by (1), ex. appertains to the exponent \u00bb mod 7 7 - . The converse of this theorem i s also true; 1. e., whatever primitive root, f> t 0 f 77~, is taken, i f c=f 41. appertains to the exponent , mod 7 7 ~ , then d is the ^greatest common divisor of -^\u00bb* \u00ab=~*L\/ . Hence i f ^ ^ \/ is the greatest common divisor ot a n d f l ' * ^ y ^ then *~~c and i^\/Tr) , The congruence (1) has, therefore, incongruent roots and these are the roots of the congruence J \/ == O j -~r^*-^-7r- . ^ Those roots of (2) which satisfy no binomial congruence of degree less than (2) are called primitive 43. roots of (2) while those roots of (2) which satisfy b i -nomial congruences of lower degree are called imprimitive roots. The primitive roots of (2) are, then, those integers which appertain to the exponent &t , mod ~t>~. They are (pfd) i n number. In par t i c u l a r , the primitive roots of 77\" are the primitive roots of the congruence If y\u00b0 i s a primitive root of (2) then the \u00b0^ roots of (2) are, by theorem 12, \/> P, f *~j J f I f o(l j are roots of the congruences and respectively, then <=^t \u00a9e i s a root of the congruence g << < \/ ^ 0 ^ ^ ^ \u00a3 T T - & In p a r t i c u l a r , i f \u00b0< and t. QTT] -fZ } then ct i s also a primitive root of 7T i n . If # is a primitive root of C\"n\\l i n 7^ ? then , ^ = \/ J i \u2022 e, 7s-\/ A Therefore, (3. === CI == \/j - ^ ^ 7 T . We must s t i l l show that CL does not appertain, mod 77\" , to an exponent less than \u00a3^W\"J . Suppose <3. appertains to the exponent f , mod 77~ , where -f- ^ ^^ir) Then and so, since t CTT] } which gives a contradiction. Therefore <2 i s a primitive root of ~7T i n ^ r ) . In order to find a primitive root of r , therefore, we need only find a primitive root of -** \u00a3V] i n . This w i l l also be a primitive root of 7T in . 45. The remaining primitive roots of 7^ may be found by applying the cor. of theorem 17. For example, to find the primitive roots of f i r s t f i n d a primitive root of -a-^ \\Z4^<>'c~\\ \u2014 \u201e To do this form the power residues of J2. , mod ^ \/ These are - V , -\/\u2022\/ * Wow form the power residues of 3 \u00bb mod *-f-l . We have so 3 appertains to the exponent f , mod \"\/\/ . -Take ^ = \u2022 r \u2022 * The greatest common multiple of , and -^ ~A i s ~~^c-=.^\/0 \u2022 Divide ^ into two factors 5w7 and prime to each other and such that i s a divisor of y and *?*cx_ a divisor of ^- . Then = s>- = ? Now Then 2^. and 3 appertain to the exponents ~ and respectively and so their product^ ~^ ' ^C^) Then the c# [_Cp(w)~\\ = ~~