4.3. Solving Special Polynomial Equations 137(1) given an arbitrary angle, to construct an angle of one third the mag-
nitude using only straightedge and compasses (‘Itisecting an Angle);
(2) given the side of a cube, to construct the side of a cube of twice the
volume (Duplication of the Cube);
(3) given the radius of a circle, to construct the side of a square whose
area is equal to that of the circle (Squaring the Circle).
After mastering the method of bisecting a general angle, many students
spend long hours trying to find a method of angle trisection and often have
trouble believing that the job is impossible-not because they are not clever
enough, but because of the intrinsic mathematical structure. Let us see
why this is so.
If there were a general trisection method, then it would work in particular
for an angle of 60 degrees. This would mean that we could construct an
angle of 20 degrees, or equivalently, construct a right-angled triangle with
hypotenuse of length 1 and one side of length cos 20’. The problem is the
following:
Given a segment of length 1, is it possible to construct a
segment of length z = cos20°, with the following operations
permitted
(1) choice of an arbitrary point
(2) construction of a straight line through two specified
points
(3) construction of a circle with a specified center and radius
(4) determination of points of intersection of two straight
lines, two circles or a line and a circle?If we introduce Cartesian coordinates in the plane, the coordinates of the
points of intersection in (4) can be found by solving linear or quadratic
equations whose coefficients lie in the smallest field determined by the co-
efficients of the equations of the lines or circles involved.
Suppose we begin with the points (0,O) and (1,0) (determining a segment
of length 1) and construct other points successively using lines and circles
determined by points already constructed. Then the coordinates of each
point so constructed would lie in some field F, which is the last in a chain
of quadratic extensions (as described in Exercise 3.9).
Can we construct the point (z,O), where t = cos 20°? Using the formula
relating cos 30 and cos 8, verify that x satisfies the equation 8x3-62- 1 = 0,
and that the polynomial on the left side is irreducible over Q. Now apply
Exercise 3.9.
Give a similar argument to show that duplication of the cube is impos-
sible.