8 CHAPTER I. 



7. Theorem. - - If two integral, functions F(x) and P(x) having 

 integral coefficients admit of no common divisor containing x modulo p, 

 p being prime, we can determine two integral functions F r (x) and P\x) 

 having integral coefficients such that 



F'(x) - F(x) -P'(x) P(x) = 1 (mod p). 

 Applying 4, we can set 



F(x) = a - A(x), P(x) = I B(x) (mod p) 



the coefficients of the highest power of x in A(x) and B(x) being 

 unity and the remaining coefficients integers. We perform the usual 

 process to determine the greatest common divisor of A and B, 

 neglecting however, multiples of p. Each remainder is congruent 

 modulo p to a product of an integer r and an integral function H(x) 

 with integral coefficients, that of the highest power of x being unity. 

 Supposing for definiteness that the degree of A is not less than that 

 of B, we obtain the congruences (mod p) : 



R m 2 = R>m lQm 4- r m . 



We derive at once the following congruences modulo p 



r,E,~A-Q,B 

 r,r,E,= -Q,A+(r,+ Q,Q,}B 

 r,r 2 r s E. d = (r 2 + Q 2 Q 9 )A- (r 2 Q + r, Q s + ft 



where M and N are integral functions of x having integral coefficients. 

 None of the integers ^ . . ., r m are divisible by p; for, A and B 

 would then have a common divisor containing x. Hence, by 4, 

 there exists an integer r such that 



r-abr^ . . .r m ^ 1 (modp). 

 From the last congruence in the above set, we therefore find 



1 = rab (MA - NB) = F(x)-rlM - P(x)-raN (modp). 



Corollary. Jf F(x) =|= [modd p, P(xJ], p being prime and P(x) 

 irreducible modulo p, we can determine an integral function F'(x) 



F(x) F(x) = 1 [modd A P(*)]. 



Note. - By an analogous use of the process for finding the 

 greatest common divisor, we obtain the following theorem: 



