158 REPORT — 1860. 



the complete solution of an Abelian congruence of order m requires only 

 two things, — 1st, the complete solution of the congruence x m — 1^0, 

 mod p, and, 2ndly, the determination of a single root of a certain con- 

 gruence of the form x m — #=0, mod p, in which a is an ordinary integer; 



so that in this case (which is that of the congruence =0, mod 43) 



we obtain a real, and not only an apparent reduction of the proposed con- 

 gruence*. 



It should also be observed that the primitive roots of the equation 



%n J 



= furnish, when rationalized, the primitive roots of the congruence 



x — 1 



%n J 



=0, mod p. This, the only direct method that has ever been suggested 



x — 1 



for the determination of a primitive root, appears to be the same as that 



referred to by Gauss in the Disq. Arith. (art. 73). 



Poinsot expresses the conviction that this method of rationalization is 

 applicable to any congruence corresponding to an equation, the roots of 

 wfaich can be expressed by radicalsf. With regard to equations of the 

 second, third, and fourth orders this is certainly true. If, for example, the 

 biquadratic equation F 4 (.t)=0 be completely resoluble when considered as 

 a congruence for the modulus p, so that F 4 (#) = (x — «j) (x — a 2 ) (x — a 3 ) 

 (or — a,), mod p, it is plain that the four roots of F(a')=0, and the four 

 numbers o„ a 2 , « 3 , a 4 may be obtained by substituting, in the general formula 

 which expresses the root of any biquadratic equation as an irrational function 

 of its coefficients, the values of the coefficients of the functions F(#) and 

 (x—a^)(x — a 2 )(# — a 3 ) (x — « 4 ) respectively. But these two sets of coeffi- 

 cients differ only by multiples of p; i. e. the values of a x , a 2 , a 3 , a t can be 

 deduced from the expressions of the roots of F(;r)=0 by adding multiples 

 of p to the numbers which enter into those expression <. But this reasoning 

 ceases to be applicable to equations of an order higher than the fourth, 

 because no general formula exists representing the roots of an equation of 

 the fifth or any higher order. If, therefore, F(.r)=0 be an equation of the 

 nth order, the roots of which can be expressed by a radical formula, and 

 which is also completely resoluble when considered as a congruence for the 

 modulus p, so that F(a,')=(.r — a t )(x — «„)... (x — a n ), mod jo, it will not 

 necessarily follow that the formula which gives the roots of F(#)=0 is also 

 capable (when we add multiples of p to the numbers contained in it) of 

 giving the roots of (x — a 1 )(x — a.,) . . . (x — a„) = 0, i. e. the roots of the con- 

 gruence F(#)=0, mod p; and thus the principle enunciated by M. Poinsot 

 is, it would seem, not rigorously demonstrated. 



67. Cubic and Biquadratic Congruences The reduction of cubic con- 

 gruences to binomial ones has been treated of by Cauchy (Exercices des 

 Mathematiques, vol. iv. p. 279), and more completely by M. Oltramare 

 (Crelle, vol. xlv. p. 314). Some cases of biquadratic congruences are also 

 considered by Cauchy in the memoir cited, p. 286. The following criteria 

 for the resolubility or irresolubility of cubic congruences include the results 

 obtained by M. Oltramare, /. c, and appear sufficiently simple to deserve 

 insertion here : — 



Let the given cubic congruence be 



* This will be at once evident, if we observe that when the congruence x m =\, modjt/, 

 is completely resoluble, its roots may be employed to replace, in Abel's method, the roots of 

 the equation x m — 1 = 0. 



t See the memoir cited above, p. 107, and M. Libri, Memoires de Mathematique et Phy- 

 sique, p. 63. 





