An introduction to the modern theory of equations, by Florian Cajori.

208

THEORY OF EQUATIONS

If in these expressions for xl, x2, X3 we substitute 0 + 2 7r for 4, the
roots undergo a cyclic permutation; that is, xl becomes x2, X2 becomes x3,
anld x becomes x1. Because of these changes, no root can, in general, be a
rational function of sin 0 and cos p; for, sin 9'
and cos 0 remaining unaltered in value when
q + 2 7r is substituted for 0q, the root could
A' undergo no change. For an arbitrary angle
i the equation I is, therefore, irreducible. Its
degree being 3, which is not an integral power
of 2, its roots cannot be constructed with the
aid of the ruler and compasses, and the trisection is impossible.
Ex. 6. Show that, if we take cos - equal to a value c, numerically < 1
and rational or involving square roots only, we get x8 = (a + iP)3, where
2 = 1 - a2, and where x = a + ip is a root which can be constructed
geometrically. Show that any number of trisectable angles 40 may be
obtained by this process. Taking a -= 2 _- /3, show that the angle
of 45~ may be trisected. By assuming a to involve at least one radical
whose order is not two nor a power of two, show how to obtain angles
which cannot be trisected.
Ex. 7. Assuming 2 cos = x, show that the trisection of the angle
depends upon the equation x8 - 3 x = 2 cos 0. Letting cos 0 = qn/n and
nx = y, derive y3 - 3 In2y = 2 in2, which has integral roots whenever the
first cubic has rational roots. If the integers m and n are prime to each
other, and n is divisible by an odd prime p but not by p2, show that ~
cannot be trisected. Prove that angles 120~, 60~,, 30~, cos- cannot be
trisected.
Ex. 8. To show that an irreducible cubic, whose coefficients are rational
numbers and whose three roots are real, cannot be solved by real radicals.
This is the so-called "irreducible case," ~ 60. We are required to prove
that in the algebraic solution of the given cubic it is impossible to avoid
the extraction of the cube root of a complex number. To this end observe,
first (~ 171, Ex. 3) that the cubic becomes a normal equation when V/D
is adjoined to 0. Here /D is real. The equation n - a = 0, where a is
not a perfect nth power, and n is prime, is irreducible. If it were possible
for the normal cubic equation to become reducible on the adjunction of
the real root X-= a, then by ~ 166, Cor. II, the degree of x"' - a = 0
would be a multiple of j, the index of the new Galois group P = 1, under