250 History of the Theory of Numbers. [Chap, viii 
then f{x) is congruent modulo p" to a product of Fi(j),. . ., F,(a:), each 
irreducible modulo p°, such that F,(x)=/,(a:)'« (mod p), where /(x)= 11/. (x)*' 
(mod p), and /.(x) is irreducible modulo p. There is an example of an 
irreducible cyclotomic function reducible with respect to every prime power 
modulus. 
P. Bachmann^^ gave an exposition of the general theory. 
G. Arnoux^"^ exhibited in the form of tables the work of finding a primi- 
tive root of the GF\1^] and of the GF[5^], and tabulated the reducible and 
irreducible congruences of degrees 1, 2, 3, modulo 5. 
S. Epsteen^°^ proved the result of Guldberg,^" and developed the theory 
of residues of hnear differential forms parallel to the theory of finite fields, 
as presented by Dickson. ^°- 
L. E. Dickson ^°^ noted that the last mentioned subjects are identical 
abstractly. Let the irreducible form A?/ be 
d'^y dy 
To the element (4) we make correspond the element 2c,z* of the Galois 
field of order p", where z is a root of the irreducible congruence 
5„2"+...+5i2+5o=0 (mod p). 
Since product relations are preserved by this correspondence, the p" resi- 
dues (4) define a field abstractly identical with our Galois field. 
Dickson^"^'' proved that x'"=.t (mod vi = p") has p and only p roots if p is 
a prime and hence does not define the Galois field of order m as occasionally 
stated. 
A. Guldberg^"^'' employed the notation of finite differences and wrote 
n m n m 
Fy, = 2 afi%, Gy, = 2 hj9%, Fy^.Gy, = 2 afi\ 2 bfi%, 
t=0 »=0 t=0 »=0 
where 6y^ = yjc+i, ^'?/x = 2/1+2. • • •> symbolically. To these linear forms with 
integral coefficients taken modulo p, a prime, we may apply EucUd's g. c. d. 
process and prove that factorization is unique. Next, let 6„, be not di\'isible 
by p, so that Gy^ is of order m. With respect to the two moduU p, Gy^, a 
complete set of p"* residues of hnear forms is a„,_i?/:r+m-i+ ■ • • +ao2/x (^1 = 0, 
1,..., p-1). Amongst these occur <}>{Gy^) =p"'{l- 1 /p'"') .. .{1-1 /p"''^) 
forms Fy^. prime to Gyj^ if Wi, . . . , niq are the orders of the irreducible factors 
of Gyx modulo p, and 
FyJ'^^'^^-^^y, (mod p, Gy,) 
In particular, if Gy^ is irreducible and of order m, 
Fyr-'^yAmodp,Gy,). 
i 
i«Niedere Zahlentheorie, 1, 1902, 363-399. 
»»Assoc. frang. av. sc, 31, 1902, II, 202-227. 
i«BuU. Amer. Math. Soc, 10, 1903-4, 23-30. 
'"/bid., pp. 30-1. 
"'"Amer. Math. Monthly, 11, 1904, 39-40. ^ 
>»"f'.\iinaU di Mat., (3), 10, 1904, 201-9. ■ 
