CHAPTER VIII. 
HIGHER CONGRUENCES. 
A Congruence of Degree n has at most n Roots if the 
Modulus p is a Prime. 
J. L. Lagrange^ proved that, if a is not divisible by the prime p, ax'^+fex""^ 
+ . . . is divisible by p for at most n integers x between p/2 and — p/2. 
For, let a, jS, . . . , p, (7 be n+1 such distinct integers. Then the quotient of 
a(a:'^-a")+6(a:'^-i-a"-^)+ . . . 
by a: — a is a polynomial aa:"~^ + . . . which is divisible by p when x=^,. . .,a. 
Proceeding as before, we finally have a{p—<7) divisible by p, which is 
impossible. 
L. Euler^ noted that a:" — 1 is divisible by a prime p for not more than n 
integers x, 0<x<p. For, if x = a, is such an integer, then x — a divides 
x^ — l—mp, where m is a suitable integer; the quotient / is of degree n — 1. 
If a: = 6 is a second such integer, x — h divides/— m'p. Proceeding as in alge- 
bra, we obtain the theorem stated. [The argument is applicable to any 
polynomial of degree n in x.] 
A. M. Legendre^ noted that P^(x — a)Q-\-pA has only one more root 
than Q. 
C. F. Gauss'* proved the theorem by assuming that there is a congruence 
ox"4- . . . = (mod p) with more than n roots a, . . ., and that every con- 
gruence of degree I, Kn, has at most I roots. Substituting y+a for x, we 
obtain a congruence a?/"-f- ... =0 with more than n roots, one of which is 
zero. Removing the factor y, we obtain a?/"~^+. . . = with more than 
w — 1 roots, contrary to hypothesis. 
Gauss^ noted that if a is a root of ^=0 (mod p), then ^ is divisible by 
x — a modulo p. li a, b,. . . are incongruent roots, ^ is divisible modulo p 
by the product (x — a){x — b).... Hence the number of roots does not 
exceed the degree of ^. 
A. Cauchy^ made the proof by use of X=(x — a)Xi (mod p), identically 
in x, where the degree of Xi is one less than the degree of X. 
A. L. Crelle^ and S. Earnshaw^ gave Lagrange's proof. 
Crelle^ proved that if ei, . . ., e„ are n distinct roots, ' 
ax^^-i- . . . = a(x — ei) . . .{x — e^+pN. 
iMem. Ac. BerUn, 24, ann6e 1768 (1770), p. 192; Oeuvres, 2, 1868, 667-9. 
='Novi Comm. Ac. Petrop., 18, 1773, p. 93; Comm. Arith., 1, 519-20. 
»M^m. Ac. Roy. Sc, Paris, 1785, 466; TWorie des nombres, 1798, 184. 
♦Disq. Arith., 1801, Art. 43. 
^Posthumous paper, Werke, 2, p. 217, Art. 338 (p. 214, Art. 333). Maser's German translation 
of Gauss' Disq. Arith., etc., 1889, p. 607 (p. 604). 
"Exercices de Math., 4, 1829, 219; Oeuvres, (2), 9, 261; Comptes Rendus Paris, 12, 1841, 
831-2; Exercices d'Analyse et de Phys. Math., 2, 1841, 1-40, Oeuvres, (2), 12. 
'BerUn Abhand., Math., 1832, p. 34. 
Cambridge Math. Jour., 2, 1841, 79. 
•BerUn Abhand., Math., 1843, 50-54. 
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