204 CHAPTER VHI. 



Suppose first that a + d =f= 0. By 201), Z becomes 



Suppose next that ft -}- y =^=Q. Applying the above procedure to 



it foUows that Zj = r . Hence Z = 0% Y M m . 



Suppose finally that a4-d = /3-t-j' = 0. Conditions 199) and 

 201) become +0_l, a/3 = 0, 



so that Z = I or M m . 



In every case, Z = Oj d or Z = 0^ Y M m . If * = 0, Ofi'sET... 

 and the theorem is proven. If A = A', A' being suitably chosen, we 



prove in the next section that every 0% is a power of L = 0^ 

 and may therefore be derived from the substitutions 195). 



203. Let Q be a primitive root of (> 2n + 1 = 1. It will satisfy an 

 equation belonging to and irreducible in the GrF[2 n ], 



p 2 +0p + l = 0. 



If we set = ^-!, Q = % m /'q m , we find that A| 2 n -h A^+ L^ m is 

 irreducible in and belongs to the GF[2 n ]. Changing the variable 

 from Q to tf = Ap, we obtain for the irreducible equation 



203) 2 +(? + A 2 ==0. 



Since the roots of 203) are tf and 2W , we have tf + 2W = 1. 

 We make the transformation of indices: 



204) 6 11 = 



Solving, we find, for p = 2, 



205) F 12 = 

 Thea 



The substitution 202) takes the form 



206) r/ 2 =rF 12 , r 3 ' 4 =r-^ 34 

 where - 



We have x 2W + 1 = 1 since (mod 2), 

 T 2_ a + (+ d)0 2n == a + (a + d)(tf + 1) - d + (cc 

 In particular, L = 0fy* takes the form 



207) ri 



