CHAPTER I, 



DEFINITION AND PROPERTIES OF FINITE FIELDS. 



1. If the difference of two integers t and r be divisible by a 

 third integer p, then t and r are said to be congruent modulo p, 

 or according to the modulus p. This property is expressed by the 

 following notation due to Gauss: 



t = r (mod p). 



For example, TEE! (mod 3), 1 = 2 (mod 5). 



The totality of integers congruent modulo p with a given posi- 

 tive integer r < p is given by the formula 



lp + r .(Z-0, 1, 2, ...). 



This totality, which will be designated C r , is said to form a class 

 of residues modulo p; it includes every integer which gives the residue r 

 when divided by p. It follows that the p classes (7 , 1? C7 2 , . . ., 

 Cp i include every integer, positive or negative. They are therefore 

 said to form a complete system of classes of residues modulo p. 



Example. - The three classes 'C , C lf C 2 form a complete 

 system of classes of residues modulo 3; indeed, every integer falls 

 under one of the three forms 3?, 3? -f 1, 3Z + 2. 



2. An instructive diagram is furnished by the regular polygon 

 of p sides inscribed in a circle. Denote the vertices taken in posi- 

 tive order (counter-clockwise) by C , C 19 . . ., C p \. Regarding (7 

 to be the origin, we take as the plot of any given integer + m that 

 vertex which is obtained by counting off from the origin m of the 

 divisions on the circle in the positive or the negative direction accor- 

 ding to the sign of m. All integers of the form Ip -f r (I = 0, 1, 

 2, . . .) are evidently plotted by the one point C r , so that congruent 

 integers give rise to the same point. The p classes of residues 

 modulo p are represented unambiguously by the p vertices of the 

 polygon. 



