Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

292

GROUP OF AN ALGEBRAIC EQUATION [CH. XIV

has the group G4. If a=b= 1, thenA = 5/4 B =5, and x4x3 +X2+X+1 =
is irreducible in R(1) and has the group C4.
Let f(x) = aoxn+axn-l... =0 be an equation with rational coefficients, irreducible in the domain of rational numbers. Prove * Exs.
10-14:
10. If there is a complex root of absolute value unity, the equation
is a reciprocal equation of even degree.
11. If there is a root r+si, where r is rational, then n is even and the
n roots may be paired so that the sum of the two of any pair is 2r, whence
r =-al/(nao). In particular, if r=0, the equation involves only even
powers of x.
12. If there is an imaginary root a+bi whose norm a2+b2 is rational,
then n is even and the n roots can be paired so that the product of any two
of a pair is a2+b2.
13. If there is a root whose absolute value p is rational, p can be
expressed in terms of the coefficients. (pn=an/ao.)
14. If we set x= py in the equation in Ex. 13, we obtain a reciprocal
equation in y.
EQUATIONS WHOSE COEFFICIENTS INVOLVE VARIABLES, ~~ 151-6
151. Definition of the Roots. We begin with the so-called general
equation (1) whose coefficients Ci,..., cn are independent complex variables. Let A(ci,..., n) be its discriminant. Let al,..., an and
Al,..., An be any two sets of constant values of cl,..., Cn for which
a0O. We shall prove that the A's can be derived from the a's by continuous variation such that, for each intermediate set of values, A X0;
expressed in geometrical language, there is a continuous path from the
point (a) to the point (A) not passing through a point of the locus A=0.
We shall prove this by induction from n-1 to n, assuming that, if
P(cl,..., Cn-) is any polynomial in ci,..., cn-, zero neither at
(li,..., ln-) nor at (L,...., L,n-), there is a continuous path from
(1) to (L) not passing through a point of the locus P=0. Neither of the
polynomials
A(C,..., C-, an) A(Cl,... Cn-, An)
is identically zero, since the first is not zero when each ci= ai, and the second
is not zero when each ci=Ai. Hence there are constants ai for which
A(cl,.., _ n-,,an)0,.. (,.., an-, An)0.
Thus there is, by hypothesis, a continuous path from (al,..., an-i, an)
to (a,..., an, an), not passing through a point of P A(ci,..
Cn -, an) = 0 and composed only of points with Cn = an, and hence not passing
* Exs. 10, 11, 13 are due to Dr. A. J. Kempner.