6 CHAPTER I. 



elements shall be the zero element u only when one of the factors 

 is U Q . Under the latter hypothesis , the series of products 



UoU;, UiUi, u-2U h ,. . ., Ug-iUi (w/H= Wo) 



are all distinct and therefore (their number s being finite) are identical 

 in some order with the series MO, Wi, Wg, . . ., ,_i. Hence if Uj be 

 any element of the set, the equation 



2) xu L = Uj (w f =4=w ) 



is satisfied by one and but one element x of the given set. Hence 

 division by any element except U Q is always possible within the set 

 and gives an unique result. 



For a field of infinite order , the assumption that division is not 

 possible in more than one way may be replaced by the above postu- 

 late that a product vanishes only when one factor vanishes. Indeed, 

 if 2) be satisfied by two distinct values x and x 2 of x, then 

 Ui(xi #2) = MO> "whereas each factor differs from U Q . 



After the above explanations, we make the formal definition: 



A set of s distinct elements forms a field of order s if the elements 

 can be combined by addition, subtraction, multiplication and division, 

 the divisor not being the element zero (necessarily in the set), these 

 operations being subject to the laws of elementary algebra, and if the 

 resulting sum, difference, product or , quotient be uniquely determined as 

 an element of the set. 1 ) 



A field may therefore be defined by the property that the rational 

 operations of algebra can be performed within the field. 



The results of 3 may now be stated in the form: The complete 

 system of classes of residues modulo p forms a field if, and only if, 

 p be a prime number. 



6. Definition of a Galois Field. Let P(x) be a rational integral 

 function of degree n having integral coefficients not all divisible by 

 a given integer p. If we divide an arbitrary integral function F(x) 

 having integral coefficients by the function P(x), we obtain a quotient 

 Q(x) and a remainder which can be written in the form f(x) +p q(x), 

 where f(x) is of the form 



3) f(x) = a Q + aix -f a 2 x 2 -\ h a n - l x n ~' L , 



each a i: belonging to the series 0, 1, 2, . . ., p 1. Then 



4) F(x) = f(x) -f p q(x) + P(x) - Q(x). 



We say that f(x) is the residue of F(x) moduli p and P(x) and write 

 A) F(x) = f(x) [modd p, P(]. 



1) Moore, Mathematical Papers, Chicago Congress of 1893, pp. 208 242; 

 Bull. Amer. Math. Soc., December, 1893. 



