DEFINITION AND PROPERTIES OF FINITE FIELDS. 9 



//' two integers f and p be relatively prime, we can determine two 

 integers f and p' sucli ilwt f'f p'p = 1. 



8. The proof of the existence of a function of degree n irreducible 

 modulo p and hence of the existence of a Galois Field of order p n , 

 for every prime p and integer n, will be given in 19 27. We 

 will first prove that no other finite fields exist and that not more 

 than one Galois Field of a given order p n exists. 



9. Consider an abstract field F[s] composed of a finite number 

 s > 1 of elements or marks u , ^l i , . . ., u s _i. Having every difference 

 u ; . Ui f the field contains a mark, denoted by M( O ), which has the 

 properties of zero viz., for every u i9 



Having every quotient 



Ui/Ui (i +=%)), 



the field contains a mark %) having the properties of unity; viz., for 



The field thus contains every integral mark 



W( C ) = w ( i) + w ( i) H ----- h w ( i) (c terms), 



((_ C )= W(0) %). 



Since there exists only a finite number of marks in the F[s], 

 there must arise equalities in the series 



...,W(_ 2) , W(-i), *( ), %), W(8), .-. 



If W( r )= w w , we have 



W(0) = W( r ) M(,) = W(r-s) - 



Denoting by p the least positive integer such that U( P ) = W(o, the p 



TT1 fit* K"S 



W(0), W(l), M(2), -, M(p_ i) 



are all distinct, while 



^(r) = W( 4 ) if, and only if, r = s (mod p). 

 This integer p is a prime number. For, if 



we have, by hypothesis, u (p j =j= U( Q ). Hence, from 



we derive W( ft ) W( ) and hence ^? 2 ^> p. Hence the integral marks of the 

 F[s] form a field F{j>] which is the abstract form of the field of the 

 classes of residues with respect to a prime modulus p. When there 

 is no ambiguity, we denote by c the integral mark w (c) . 



