DEFINITION AND PROPERTIES OF FINITE FIELDS. 13 



such marks. Each mark w belonging to the exponent s 1 is called 

 a primitive root of the equation 



and also a primitive root of the F[s]. Since the powers w 1 , w 2 , . . ., itf~ 1 

 are all distinct, we may state the theorem: 



Ttie p n \ marks =j= of the F[s = jp n ] are the p n 1 successive 

 powers of a primitive root of that field. 



Corollary. If d be any divisor of p n 1, the mark w& n ~V/ d belongs 

 to the exponent d. 



18. We may now recognize in our F[s] the abstract form of a 

 Galois Field of order s p n . Indeed, by 11, the primitive root w 

 satisfies an equation of degree ~k < n. 



W k (x) = 0, 



belonging to and irreducible in the JF[jp]. Every mark =(= of 

 the F[s], being a power of w, can be reduced by the identity 

 W k (w) to the form 



where the c's belong to the -F[j>]. The mark zero evidently falls 

 under this form. Since, inversely, every one of these p k expressions 

 is a mark of the F\s\, we must have k = n. Hence every mark of 

 the F[s =jp rt ] represents a class of residues moduli^), a prime, and W n (x\ 

 a function with integral coefficients irreducible modulo p. Every 

 existent field is therefore the abstract form of a Galois Field. 



Suppose there could exist a second field F'{j) ri ] of order equal 

 to that of F\jp n ~\. The field -F[j) n ] possesses a primitive root w 

 satisfying an equation W n (x) = 0, of degree n, belonging to and 

 irreducible in the -F[jp]. The function W n (x) divides x pn x in 

 the F[p] 1 }. We may, indeed, apply in the F[p] Euclid's process 

 for finding the greatest common divisor of these functions. If there 

 were no common factor, we would ultimately reach as a remainder 

 a constant, whereas the process may be interpreted in the 6rjP[j? n ], 

 in which field the common factor x w exists. Hence W n and 

 x pn x have a common factor in the F[p~\. Moreover, W n is irre- 

 ducible in that field. 



Since F[p] is contained in -F"[j> w ], the division of x pn x by 

 W n is, a fortiori, possible in the .F'jj/ 1 ]. It follows from 13 that 



i 

 1) Another proof is given in 23. 



