ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 63 



Hence , for p =j= 2, we must have /3 = 0. The power m -f 2 of 

 53) f+| 3 + y | 



requires C m+8>i y + CL,+i,8-3'y + CL+8.4* 4 - 0. 



If jp is neither 2 nor 7, O m + 2j2 =|= (mod p) and may be divided out; 

 for, if m -f- 2 be divisible by ^ then is also 5(m -f- 2) =^) n + 7. 

 Multiplying the resulting equation by 5 2 and replacing 5m and 5(m 1) 

 by ^? w 3 and p n 8, respectively, we have for p =)= 2, p =f= 7, 



25y 2 - 15y -f 2 4 EE(5^ - 2 )(5y - 2 2 ) = 0. 

 The power m -f 4 of 53) requires, if 1 ) ^> n > 13, 



On+4.,55(^M-^+4,620y^ 



If p is not 2, 3, 7 or 17, we may divide out the factor 



(m + 4) (m -f 3) (m -f 2) (m + l)m. 



Multiplying afterwards by 5 4 7! and replacing 5(m 1) by 8 (mod^), 

 etc., we find 



This equation is an identity for by = a 2 , but reduces to 10 a 9 for 

 by = 2 a 2 . In the latter case, a = y = 0, if p =j= 2. Hence, for 

 #*=)= 13, 2", 3", 7 W or 17 W , the only possible quintic which represents 

 a substitution on the marks of the GF[p n = 5m -f 3] is reducible to 



5 5 + 5a 3 -j- 2 . 



We have shown in 80 that this quintic is indeed a SQ[5,p n = 5m -f- 3], 

 The special cases above excluded require separate treatment. 



f 90. The foregoing methods may be employed 2 ) to show that the 

 following table gives every reduced SQ[]k, p"] for Jc < 6: 



Beduced quantic Suitable for p n = 



............... any p* 



....... ........ 2" 



5 s a | (a = not -square) ...... 3" 



J 4 3| ............. 7 



|4 + a s | -f a s g (if it vanishes only for g 0) 2" 



| 5 ............... 5", 5m 2, 5m -f 4 



| 5 ag (a not a fourth power) . . . . 5 n 



............ 32 



............ 7 



2 +3 2 (a- = not -square) . 7 



1) If #=13, the power w-f-4 = 6 brings in terms |84= |2(/m 1). 



2) Compare the author's Dissertation, I.e. pp. 77 86 and 101102. 



