236 History of the Theory of Numbers. [Chap, viii 
congruences modulo p are of the form a-}-h\/n, where n is a fixed quadratic 
non-residue of p, while a, b are integers. But the cube root of a non-cubic 
residue is not reducible to this form a-\-hy/n. The p+1 sets of integral 
solutions of y^ — nz'^=a (mod p) yield the p-|-l real or imaginary roots 
x^y-[-zy/n of x^^=a (mod p). The latter congruence has primitive roots 
if = 1. 
Th. Schonemann^ built a theory of congruences without the use of 
Euclid's g. c. d. process. He began wath a proof by induction that if a 
function is irreducible modulo p and divides a product AB modulo p, it 
divides A or B. ■Much use is made of the concept norm NJ^ of f{x) with 
respect to <i>{x), i. e., the product /(j8i) . . ./(i3^), where i3i,. . ., jS^ are the 
roots of 4>{x)=0; the norm is thus essentially the resultant of / and 0. 
The norm of an irreducible function with respect to a function of lower 
degree is shown by induction to be not divisible by p. Hence if / is irre- 
ducible and Nf^=0 (mod p), then/ is a divisor of modulo p. A long dis- 
cussion shows that if ai, . . . , a„ are the roots of an algebraic equation 
/(x)=a:"-f . . . =0 and if /(a:) is irreducible modulo p, then niii]z— 0(a,)[ 
is a power of an irreducible function modulo p. 
If a is a root of /(x) and f{x) is irreducible modulo p, and if 4>{a) 
=^(a)-f-pi?(a), we write (p^^/ (mod p, a); then (f>{x)—\p{x) is divisible 
by J{x) modulo p. If the product of two functions of a is =0 (mod p, a), 
one of the functions is =0. 
If /(x) =x'*-f ... is irreducible modulo p and if /(a) =0, then 
/(a;) = (x-a)(x-a''). . .(x-a''""'), aP"-^=l (mod p, a), 
n-l P"~' 
x^ — 1 = n )x— 0,(a)|- (mod p, a), 
where 4>i is a polynomial of degree n — 1 in a with coefficients chosen from 
0, 1,. . ., p — 1, such that not all are zero. There exist (^(p" — 1) primitive 
roots moduhs p, a, i. e., functions of a belonging to the exponent p" — 1. 
Let F{x) be irreducible modulis p, a, i. e., have no divisor of degree ^ 1 
modulis p, a. Let F{^) = 0, algebraically. Two functions of ^ with coeffi- 
cients involving a are called congruent modulis p, a, j3 if their difference is 
the product of p by a polynomial in a, /3. It is proved that 
F{x)^ix-^){x-^n . ..{x-^^'"'-'"'), /3^'""^1 (mod p, a, ^). 
If v<n, n being the degree of f(x), and if the function whose roots are 
the (p"— l)th powers of the roots of /(x) is ^0 (mod p) for x = l, then /(a;) 
is irreducible modulo p. Hence if ??? is a divisor of p — 1 and if g^ is a primitive 
root of p, and if k is prime to m, then x"* — ^* is irreducible modulo p. 
If p<m, m being the degree of F{x), and if the function whose roots are 
the (p*^— l)th powers of the roots of F{x) is ^0 (mod p, a) for x = l, then 
"Grundziige einer allgemeinen Theorie der hohem Congruenzen, deren Modul eine reelle 
Primzahl ist, Progr., Brandenburg, 1844. Same in Jour, fiir Math., 31, 1846, 269-325. 
