DEFINITION AND PROPERTIES OF FINITE FIELDS. 5 



5. Definition of a field. A set of elements u lt u 2 , . . ., u a , which 

 may he combined by addition subject to the formal laws 



such that the sum of any two elements is likewise an element of 

 the set is called an additive- field. If two elements Ui and u k are 

 given, there may or may not exist a third element HJ in the set such 

 that Ui + Uj = u k . If existent , HJ is said to be determined by sub- 

 traction, Uj EE Uk Ui. Assume 1 ) that subtraction is always possible 

 in the given additive -field. The set will contain the differences 

 MX Mi,"M2-- M2, ''> u a u . Each has the additive property of 

 zero, since % -f (M/ M/) = My. From the latter, M, : M/ = M/ */ 

 follows by the definition of subtraction. Hence the above differences 

 all have a common value u. There exists no new zero element M f , 

 since Uj + u' = Uj requires u' = Uj % = M. Two elements are called 

 equal or distinct according as their difference is or is not the zero 

 element u. Select from the original set all the distinct elements^and 

 denote them by M O , MI, Ma, . . ., 'M,_I, where M O denotes the unique 

 zero element. 



Assume next that the s elements MO, MI, . . ., M,_ i may be com- 

 bined by multiplication subject to the formal laws 



UiUj = M;M/, Ut(UjU k ) = (UtUj) Uk, M^M; M*) = MfM; M/Mjfe, 



P 



such that the product of any two elements is itself an element of 

 the set. Then the element M O will have the multiplicative properties 

 of zero, viz., for any element Uj of the set, 



Indeed, since every product M/MI is an element of the set, 



Uj(U; U^ = M/M; MyM/ = MQ, (M/ M,-)w/ = M . 



Given two elements u- t and M*, M/=J=M O , there may or may not 

 exist a third element Uj in the set such that M,M, = M A . If existent, 

 Uj is said to be determined by division, UjUk/Uf. Assume 2 ) lastly 

 that division is always possible in the set, and in a single way, the 

 divisor being other than the zero element. A set of s distinct 

 elements satisfying the above four conditions is said to form a field 

 of order s. 



To obtain a field of finite order, the assumption concerning 

 division may be replaced by the postulate that a product of two 



1) In the additive -field of all positive integers, not every difference of 

 two elements belongs to the field. 



2) The set of all positive and negative integers satisfies the assumptions 

 as to addition, subtraction and multiplication, but not that for division. 



