248 History of the Theory of Numbers. [Chap, viii 
Analogous to the definition of trigonometric functions in terms of expo- 
nentials, he defined quasi cosines and sines by 
and Tqx as their quotient. Their relations are discussed. He defined 
pseudo cosines and sines by 
Cpx = Cq[{p - l)x] = e-'Cq2x, Spx = -e-'Sq2x. 
For each prime p<100, he gave (pp. 65-86) the (integral) values of 
e^, ind x, Cqx, Sqx, Tqx, Cpx, Spx 
for x=l, 2,. . ., p + 1. 
L. E. Dickson^^ extended the results of Serret^* to the more general case 
in which the coefficients of the functions are poljTiomials in a given Galois 
imaginary (i. e., are in a Galois field of order p"). For the corresponding 
generaUzation of the results of Serret^^ on irreducible congruences modulo 
p of degree a power of p, additional developments were necessary. To 
obtain the irreducible functions of degree p in the GFlp'^'] which are of the 
first class, we need the complete factorization, in the field, 
hiz^'-z-v) =U{h''z''-hz-^) 
where hv is an integer and /S ranges over the roots of 
all of whose roots are in the field. For the case v = this factorization is 
due to ]Mathieu." Thus K^z^—hz—^ is irreducible in the field if and only 
if B^O. In particular, if /3 is an integer not divisible by p, z^ — z—^ is 
irreducible in the GF[p"] if and only if n is not divisible by p. 
R. Le Vavasseur^^ employed Galois imaginaries to express in brief no- 
tation the groups of isomorphisms of certain tj-pes of groups, for example, 
that of the abelian group G generated by n independent operators ai, . . ., 
a„, each of period a prime p. If i is a root of an irreducible congruence 
of degree n modulo p, and if j = ai-\-ia2+ . . .+i''~^a„, he defined a^ to be 
Oi"' . . . an"". Then the operators of G are represented by the real and 
imaginary powers of a. 
A. Guldberg^^ considered linear differential forms 
A d^y , , dy , 
^y=^>:d^.-^----^^^di+^oy, 
^4th integral coefficients. The product of two such forms is defined by 
Boole's sjTnbolic method to be 
d'' 6} d 
Ay'By={au-^-\- • • • +«o)(^/^+ • • • +^^+^o)2/. 
"BuU. Amer. Math. Soc.,3, 1896-7, 381-9. 
"M^m. Ac. Sc. Toulouse, (9), 9, 1897, 247-256. 
"^Comptes Rendus Paris, 125, 1897, 489. 
