VIII PREFACE. 
X<n; taking X= 1, we see that, if G is not divisible by p^, g' is a primitive 
root of p^ and of all higher powers of p. This Chapter VII presents many 
more theorems on exponents, primitive roots, and binomial congruences, and 
cites various lists of primitive roots of primes < 10000. 
Lagrange proved easily that a congruence of degree n has at most n roots 
if the modulus is a prime. Lebesgue found the number of sets of solutions of 
01^1"*+ • • • -\-akXk"=a (mod p), when p is a prime such that p — 1 is divisible 
by m. Konig (p. 226) employed a cyclic determinant and its minors to find 
the exact number of real roots of any congruence in one unknown; Gegen- 
bauer (p. 228) and Rados (p. 233) gave generalizations to congruences in 
several unknowns. 
Galois's introduction of imaginary roots of congruences has not only 
led to an important extension of the theory of numbers, but has given rise 
to wide generalizations of theorems which had been obtained in subjects 
like linear congruence groups by applying the ordinary theory of numbers. 
Instead of the residues of integers modulo p, let us consider the residues of 
polynomials in a variable x with integral coefficients with respect to two 
moduH, one being a prime p and the other a polynomial f{x) of degree n 
which is irreducible modulo p. The residues are the p" polynomials in x of 
degree n — 1 whose coefficients are chosen from the set 0, 1, . . . , p — 1 . These 
residues form a Galois field within which can be performed addition, sub- 
traction, multiplication, and division (except by zero) . As a generahzation 
of Fermat's theorem, Galois proved that the power p" — 1 of any residue 
except zero is congruent to unity with respect to our pair of moduli p and 
f{x). He avoided our second modulus f{x) by introducing an undefined 
imaginary root i of f{x) = (mod p) and considering the residues modulo p 
of polynomials in i; but the above use of the two moduH affords the only 
logical basis of the theory. In view of the fullness of the reports in the text 
(pp. 233-252) of the papers on this subject, further comments here are 
unnecessary. The final topics of this long Chapter VIII are cubic congru- 
ences and miscellaneous results on congruences and possess little general 
interest. 
In Chapter IX are given Legendre's expression for the exponent of the 
highest power of a prime p which divides the factorial 1-2. . .m, and the 
generalization to the product of any integers in arithmetical progression; 
many theorems on the divisibility of one product of factorials by another 
product and on the residues of multinomial coefficients ; various determina- 
tions of the sign in 1-2... (p — l)/2==tl (mod p); and miscellaneous 
congruences involving factorials. 
In the extensive Chapter X are given many theorems and formulas 
concerning the sum of the kth. powers of all the divisors of n, or of its even or 
odd divisors, or of its divisors which are exact sth powers, or of those divisors 
