Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 237 
F{x) is irreducible modulis p, a. Hence if m is a divisor of p"— 1, and if 
^(a) is a primitive root of 
a;^"-^=l (mod p, a), 
and if k is prime to m, then x^ — g^ is irreducible modulis p, a. 
If F(x, a) is irreducible modulis p, a, and if at least one coefficient satisfies 
^p'-i^j (mod p, a) 
if and only if j' is a multiple of n, then 
1^(0^)= n F(a:, a^) (mod p, a) 
y=o 
has integral coefficients and is irreducible modulo p. 
If G{x) is of degree mn and is irreducible modulo p, and G(a) = 0, alge- 
braically, and if ^(a) is a primitive root of a;^'"''=l (mod p, a), then 
X(a:)^n (x-F), < = r^ 6 = ^-^, 
y=o p — 1 
has integral coefficients and is irreducible modulo p. 
The last two theorems enable us to prove the existence of irreducible 
congruences modulo p of any degree. First, 
(x'"-"-'-l)/{x'"'"'-'-l) 
is the product of the irreducible functions of degree p" modulo p. To prove 
the existence of an irreducible function of degree Zp", where I is any integer 
prime to p, assume that there exists an irreducible function of each degree 
<Zp", and hence for the degree a = ylp", where A=(f){l)<l. Let a be a 
root of the latter, and r a primitive root of x^~^= 1 (mod p, a), where P = p'^. 
Since I divides P — 1 by Euler's generalization of Fermat's theorem, x^ — r 
is irreducible modulis p, a. Hence by the theorem preceding the last, 
JI]Zq{x^ —r'^) is irreducible modulo p. Since its degree is Ip^'A, the last 
theorem gives an irreducible congruence of degree Zp". 
Every irreducible factor modulo p of x^"~^ — 1 is of degree a divisor of n. 
Conversely, every irreducible function of degree a divisor of n is a factor 
of that binomial. If n is a prime, the number of irreducible functions 
modulo p of degree n" is (p"'— p""^" )/n\ If n is a product of powers of 
distinct primes A, B,. . ., say four, the number of irreducible congruences 
of degree n modulo p is 
_ipABCD_pABC_ _pBCD\pABi A.pCD_pA_ _pD] 
Tv 
where p = p"/(^sc'Z)) Replacing p by p"*, we get the number of irreducible 
congruences of degree n modulis p, a, where a is a root of an irreducible 
congruence of degree m. 
If n is a prime and p belongs to the exponent e modulo n,/= {x'' — \)/{x — \) 
is congruent modulo p to the product of (n — l)/e irreducible functions of 
