DEFINITION AND PROPERTIES OF FINITE FIELDS. 7 



The totality of functions F(x) obtained by giving to the poly- 

 nomials Q(x) and q(x) in 4) all possible forms is said to constitute 

 a class of residues; two functions are called congruent if, and only 

 if, they belong to the same class of residues. From the form of 3) 

 there are evidently p n distinct classes. 



Consider two integral functions having integral coefficients 



Fi(x) - /,< +p-q, (as) + P(x) Qi (x) [ = 1 ., 2]. 



It is evident that the class to which F 1 + F 2 or F t F 2 belongs 

 depends merely upon the functions f l + f 2 or /j/g respectively, being 

 independent of the functions q if Qi. Hence classes of residues com- 

 bine unambiguously under addition, subtraction and multiplication. 

 In order that the division of an arbitrary class by any class (7, not 

 the class zero (7 , shall lead uniquely to a third class, it is necessary 

 that the equation d = Co shall require d = Co- Evidently this 

 will not be the case if p be composite, p=Pip 2 , or if P(x) be 

 reducible modulo p, viz., 



P(x)-P t (x)P t (x)+pP t (x) 



where the P(#) are integral functions having integral coefficients, 

 the degrees of P^ (x) and P 2 (x) being less than the degree of P(x). 

 Hence p must be prime and P(x) irreducible modulo p. 



Inversely, if p be prime and P(x) irreducible modulo p, it 

 follows from 7 that to any class C Fl other than the class C there 

 corresponds an unique class CF\ such that Cf^Cp^ is the class unity. 

 Hence there exists the quotient class 



The p n classes of residues therefore form a field called a Galois Field 

 of order p n . Moreover, the p n classes of residues moduli p and P(x) 

 form a field if, and only if, p be prime and P(x) be irreducible 

 modulo p. 



As an example, let p = 3 and P(x) = x 2 x 1. The 3 2 resi- 

 dues are 



0, 1, -1, x 9 x + 1, x1, x, -#+1, -x1. 



The sum, difference or product of any two of these may evidently 

 be reduced moduli 3 and x 2 x 1 to one of the nine residues. 

 Moreover, the quotient of any one by any residue except may be 

 reduced to one of the set. For example, 



- 1 * x a* 2 _ 1 



= L > ^ == = 



The nine residues thus form a Galois Field of order 3 2 . 



