DEFINITION AND PROPERTIES OF FINITE FIELDS. 

 i2. Let u be any mark =f= of the F[s = p*\. The marks 



belonging to our finite field are not all distinct. From u r = u s , we 

 derive u r ~ s =^\. The least positive integer e for which u e = 1 is 

 called the period of the mark u, while it is said to belong to the 

 exponent e. The marks 1, u, u 2 , . . ., w e1 are all distinct. 



We may form a rectangular array of the marks =4= of the field 



as follows: 



1 u u* . . . u*~ 



. . . U e 



where ^ is any mark =j= not occurring in the first line, u 2 any 

 mark =)= not in the first or second lines, etc. Evidently the marks 

 in any line are different from each other and from those in the 

 preceding lines. Since each new mark HI gives rise to a set of e 

 new marks, the number p n 1 of the marks =(= in the F[p n ~\ is a 

 multiple of e. 



Theorem. The period of any mark =)= of the JP[p w ] is a divisor 

 of p n 1. 



13. Raising u" to the power (p n l)/e, we have 

 .^ w -i=l, if u 4=0. 



We have thus the following generalization of Format's Theorem: 

 Every mark of the -F[jp w ] satisfies the equation 



We have therefore the following decomposition in the 



t = 



HI running over the p n marks of the 



14. Theorem. - - // two marks tt L ,u 2 belong respectively to ex- 

 ponents e^ which are relatively prime, their product u^ belongs to the 

 exponent e L e 2 and the e e 2 marks 



are all distinct. 



