MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 53 



hence belongs to the GF[p n ]. Hence the m conjugate marks are the 

 roots of an equation of degree m with coefficients in the GF[p n ]. 



By 31, the roots of an equation of degree m belonging to and 

 irreducible in the GF[p n ] are conjugate with respect to the GF[p n ]. 



In particular, the marks A, A pn of the GF[p Zn ~\ are conjugate 

 with respect to the GF[p n ]. The conjugate A pH of A will be 

 designated by A. Evidently A = A if, and only if, A belongs to 

 the GF[p n ]. The following relations are proven at once: 



A B = A-B = BA, A = ', + B = A + B, (A/B) = A/B. 



74. Neivtoris identities. If S ( denote the sum of the t ih powers 

 of the roots of the equation 



f(x) = x + a^-i-h a 2 #'"- 2 -f .+ 0*1-1 a + a m = 0, 

 in which the coefficients a, Mong to the GF[p n ], then 



= !+&! 

 = 



40) = 



= m _i + aiS m -2+ a 2 ^ m _ 3 H ----- h-2^i4-(w l)a m _r 



These identities follow as in algebra upon equating the coefficients 

 of like powers of x in the following identity, in which cc v . . ., a m 

 are the roots of f(x) = 0: 



41) 



This identity, evidently true for.m = l, may be proven by 

 simple induction 1 ). Supposing it true for a particular w, we have 

 proven it true for the value m -f- 1. Let 



F(x)=f(x)'(x a m ^i)=a? n + 1 +(ai a m+ i)x m +(a 2 ^im-fi)^ m ~H 



+ (m Urn lm-f l) % a m (X m + i 



Multiplying 41) by x a w +i and adding /"(ic) to the left member 

 and x m -\ h o>m to te right member, we find 



^ -| ^ = (m + 1 ) # m + wYa-i a m 4. i) ic m ~~ 1 



XCti X ttm-l-l 



= 1 



-f (m - 



m - 1 -f i). 



1) Since equations in the GF[p*~\ are not algebraic identities, we avoid 

 the consideration of derivatives. We might, however, employ Weber's definition 

 (Algebra, I, 13) of the derivatives of a polynomial in x for the derivatives up 

 to the jpth, but not for the higher derivatives on account of the denominators TI(m). 



