56 
ALGEBRAIC INVARIANTS 
8. If «i, a 2 are the roots of the binary quadratic form /, and c* 3 , a 4 the 
roots of/' in § 11, the simultaneous invariant 
ac' -f a'c—2bb' = aa! {a&n +«№ — |(«i +«2) («3 +«4) | = %a 0 {u—v), 
if the product ff is identified with the quartic in Ex. 7. Hence a simul 
taneous invariant of the quadratic factors of a quartic is an irrational invar 
iant of the quartic. Why a priori is the invariant three-valued? 
9. The cross-ratios of the four roots of the quartic are — v/u, etc. These 
six are equal in sets of three if 7 = 0. For, if s = 0, 
vw=u\—v—w) — u 2 , UW—V\ — u—w) = z> 2 , —=—= . 
u w v 
The remaining three are the reciprocals of these and are equal. 
10. By Ex. 3, § 11, one of the cross-ratios is —1 if ac'+ . . . =0. Why 
does this agree with Ex. 8? 
11. The product of the squares of the differences of the roots of the 
cubic equation in Ex. 7 is known * to be 
— 4(—12/) 3 —27S 2 =ao 6 (t{ — v) 2 (u — w) 2 (v~ w) 2 . 
Also,* 5 2 = 256(/ 3 —27/ 2 ). Hence the left member becomes 3 6, 4 4 / 2 . Thus 
33.42/= ±a„ 3 (M—v)(u—w){v—w). 
Using J from § 31, and the special values in Ex. 7, show that the sign is 
plus. Verify that the cross-ratios equal —1, —1., 2, 2, if 7 = 0. 
36. Covariants in Terms of the Roots. Let K (oq, . .. ,a p ;x, y) 
be a covariant of constant degree d (in the coefficients) and 
constant order co (in the variables) of the binary form/=aox p + ... 
where k is a polynomial in x/y and the roots ai, . . . , a v of 
/= 0. Under the transformation T„ in § 28, let / become 
Aq£ p + . . . , with the roots a 1, . . . , a v . Then 
y V 
Making use of the identities 
<Xi= (d!i —q:i)+q;i 
* Cf. Dickson, Elementary Theory of Equations, p. 33, p. 42, Ex. 7.