ON THE THEORY OP NUMBERS. 157 



and attending to the limitation to which the cube roots are subject, 



ar==— 8, +11, +21, or, -2,+4, +16; mod 43. 



Thus the complete solution of a congruence of the sixth order is obtained by 

 means of binomial congruences of the second and third orders only. 



An essential limitation to the usefulness of this method arises from the cir- 

 cumstance that it does not always (or even in general) happen that (as in 

 the example just given) each surd entering into the expression of the root 

 becomes separately rational. For that expression may itself acquire a rational 

 value, while certain surds contained in it continue irrational, precisely as, in 

 the irreducible case of cubic equations, a real quantity is represented by an 

 imaginary formula. To illustrate this point by an example, let us consider 



^ i 



the same congruence =0 with respect to the modulus 29| Here in 



x—l 



the expression 

 •1+ V^7 



x=- 



where p denotes a cube root of unity, we have, putting V— 7^ +14, and. 

 P = l, 



the irrational cube roots disappearing of themselves. Again, putting 



P=k-l±V-3), 



we find 



*=7± 3 



iv^(|v2i)w+(^y 



~7 + (7)*=7±16 = -6 or -9, 

 where every radical becomes rational of itself. Similarly taking the values 

 V^7 = — 14, p=o( — !± ^ — 3). we find x = — 5 or — J3. But lastly, 

 putting V^7= — 14, p=l, we find 



x=]2 + g [14 + 7 V«]*+g [14-7^2]*. 



To rationalize this expression, we have to observe that 14 + 7 V2, relatively 

 to the modulus 29, is the cube of a complex number of similar form ; in fact, 

 we have (14±7 V2)=(5±ll V2) 3 , mod 29, whence x=— 4. To elicit, 

 therefore, the value of this root from the irrational formula, we are obliged to 

 solve the cubic congruence # 3 = 14 + 7 V2, which, although of lower dimen- 

 sions than the proposed congruence, is probably less easy to solve tentatively, 

 because 29 has 29" — 1 =840 residues of the form a + b V2, and only 29 — 1 

 = 28 ordinary integral residues; so that practically the method fails. Theo- 

 retically, however, the relation between the analytical expression of the 

 equation-roots and the values of the congruence-roots is of considerable 

 importance, and the subject would certainly repay a closer examination 

 than it has yet received. We may add that, if m be a divisor of p — 1, 



% Ibid. p. 132. 



