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X<n; taking X«l, we see that, if G is not divisible by p2, g is a primitive root of p2 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 ajXi*».|_ .. +akXkm~a (mod p), when p is a prime such that p — 1 is divisible by w. 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 polynmnials in a variable x with integral coefficients with respect to two moduli, one being a prime p and the other a polynomial f(x) of degree n which is irreducible modulo p. The residues are the pn polynomials in a? 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, subtraction, multiplication, and division (except by zero). As a generalization of Fermat's theorem, Galois proved that the power pn —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 /(#) by introducing an undefined imaginary root i of /(a;)==0 (mod p) and considering the residues modulo p of polynomials in i; but the above use of the two moduli 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 congruences 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.. .w, 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 determinations of the sign in 1-2... (p — l)/2ss=*=l (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