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

282

GROUP OF AN ALGEBRAIC EQUATION [Cn. XIV

EXERCISES
1. If x3+cx2+dx+e=O, where c, d and e are integers, has a rational
root, that root is an integer. Hint: Let x= a/b, where a and b are relatively
prime integers, and multiply the equation obtained from the cubic equation
by b2.
2. An integral root of the equation in Ex. 1 is a divisor of e. Hint:
It divides x3, cx2 and dx.
3. X3-3x+l is irreducible in R(1).
4. x3-7x+7 is irreducible in R(1).
5. State and prove for equations of degree n the theorems corresponding to those of Exs. 1, 2 for n=3.
6. x4+x3'x2+x+1=O is irreducible in R(1). Hints: It has no
rational root (Ex. 5). If it has the factors x —+ax+r, x2+bx+r-, where
a, b, r are rational, then
a+b=l, ab+r+r-'=l, ar-'+br=l.
Either a= (1 lV/5), b=(lF x5), r=1;
r 1
or a= b= — r4 +rr3+r2+r+1=O.
r+l1 r+l
145. Functions with n! values. Let R be a given domain
which contains all of the coefficients of a given numerical
equation
(1) f(x)-x -ClXn-+C2Xn-2-... +(-l)n C=0,
which, without * real loss of generality, will be assumed to have
the distinct roots xi,...,. There exist integers mi,...,
mn such that
V1=mIXl +m2X2+..~ ~ +-mXn
gives rise to n! numerically distinct functions Vs when the
n! substitutions s on xi,..., x, are applied to it. For, if
s and s' are different substitutions, Vs and Vs, are not equal
identically as to mi,..., m,. We can, however, choose
integers mi,..., mn, which satisfy no one of the n!(n!-1)/2
equations of the form Vs =Vs,. In fact, ml=m2 is the only
one of these equations involving only ml and m2. Give to
ml any integral value (say 0) and to m2 any integral value
* For, if it has a multiple root, f(x) and its derivative f'(x) have a greatest
common divisor g(x) with coefficients in R. Then f(x)/g(x) has its coefficients
in R, has no multiple root, and vanishes for each root of f(x)=0.