44 CHAPTER IV. 



Ex. 12. (Mathieu). If H belong to the GH[p nm ], we have the, 

 decomposition 



where ^ runs through the series of marks of the GF[p n \. 



60. Table of primitive irreducible quantics 1 ). When more than 

 one PIQ\m, p] is known, we choose that one x m ~ ax' '-f fix'' 1 -j- . . . 

 (mod p) in which the exponent r is as small as possible. 

 Modulo 2: x 2 :^x+l, afea-fl, x^x+l, x=x*-\-l, x=x + l, 



Modulo 3: x 2 =2x+l, x*=x + 2, x*=2 x*+2x 2 + x+1, x*EE 

 Modulo 5 : x 2 = 2x + 2, x*~ 2x + 3, x*= x 2 + x + 2, x*= x + 2, 



Modulo 2 ) 7 :x*=x-3, x*=x-2, x*=2x*+2x + 2, x=6x+3, 



x= - x- x*- x 5 - x 2 - x - 3, x 1 = x + 3. 

 Modulo 11: x 2 ~4x-2. 



CHAPTER IV. 



MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 

 Squares, not-squares, m ih poivers in a Galois Field, 61 63. 

 61. Every mark of the GF[2 n '\ satisfies the equation x 2 = x, so 



that x is the square of the mark x 2 . Every mark has one and 

 only one square root, since 1 = -j- 1 in the 6rjF[2 re ]. 



In the GF[p n ], p > 2, a mark may or may not be the square 

 of a mark belonging to the field, and is called a square or a not- 

 square respectively. If p be a primitive root of the GrF[p n ], so that 



36) ^"- 1 = 1, 0(^-D/ 2 = - 1, 



the even powers of p are squares, 2/ '= (+ p 7 ') 2 ; while the odd powers 

 are not-squares. In fact, ^ 2/t + 1 = x 2 would require 



0(2A+l)(p*-l)/2^pA(p-l) . Q(p n -lV* = __! = xP n ~ l = +1. 



Hence there are (p n I)/ 2 squares and as many not-squares in the 

 GrF\p n ~\. Furthermore, the product or quotient of two squares or of 

 two not-squares is again a square; but the product or quotient of a 

 square by a not -square, or vice versa, is a not -square. 



1) A table of irreducible quantics (not all primitive) is given by Jordan, 

 Comptes Eendus, 72 (1871), pp. 283 290. His quantic # 8 -f x s + x*+x + 1 is 

 divisible by # 3 -f # 2 -|- 1 modulo 2, while # 8 -j-#-f2 is divisible by x 5 mod 11. 



2) Serret, Cours d'Algebre superieure, II, pp. 181 189. 



