OF THE HIGHER DEGREES, WITH APPLICATIONS. 121 
THe AUXILIARY BiQUADRATIC WITH A QUADRATIC SuB-AUXILIARY. 
§64. Proposition XXIII. In order that 7;, taken as in (62), 
may be the root of an irreducible equation / (~) = O of the fifth 
degree, whose auxiliary biquadratic has a quadratic sub-auxiliary, it 
must be of the form 
1 1 2 2 
mae (4b +42) + (a AP ay Ab) } (65) 
where 4, and 42 are the roots of the irreducible equation 
dy (ve) =a? —2 pert+q=—0;anda =~b+dVY (p—?q), 
a@=b—d / (p?— qq); p, 6 and d being rational ; and the 
1 1 1 at 
roots A and ds being so related that Ae Ag ee 
By Prop. VII., when a quintic equation is of the first (see §30) 
class, the auxiliary biquadratic is irreducible. Hence, in the case 
we are considering, the quintic is of the second class. The quadratic 
sub-auxiliary may be assumed to be ¢; (x) = a? — 2 px +k = 0, 
p and & being rational. By Prop. XXI., the roots of the equation 
¢ (x) = 0 are.4, and hi A . Therefore k = (Ay 4 )® ; or, putting 
g for hi 45, k = g®. By the same proposition, fy; 4; is rational. 
Therefore g is rational. Hence ¢; (x) has the form specified in the 
enunciation of the proposition. Next, by Proposition XVI., there is 
‘a 4 
a fifth root of unity ¢ such that ¢ Ae =e Ay . If we take ¢ to be 
unity, which we may do by a suitable interpretation of the symbol 
1 N 4 3 2g 
Ap , 4s =m 4? . This implies that e 4? = a, 43 , a2 being 
what a becomes in passing from 4; to 42. Substituting these values 
3 S 
of e, 4; and hy A? in (62), we obtain the form of 7; in (65), while at 
‘al 
the same time ae We = h; 4) = q. The forms of a and ag have to 
1 
be more accurately determined. By Prop. XIV., 4; is the only 
principal surd that 7, as presented in (62), contains. Therefore 
a involves no surd that does not occur in 4; ; that is to say, 
/ (p? — @g®) is the only surd in a. Hence we may put 
mM™=b+d/(p?—¢);bandd being rational. But ag is what 
a becomes in passing from 4; to 42. And 4p differs from 4; only 
in the sign of the root / (p? — q°). Therefore 
a, =b—d/(p?— q’). 
§65. Any rational values that may be assigned to 6, d, p and q 
in 7, , taken as in (65), make 7; the root of a rational equation of the 
