ABSTRACT
This section presents three famous compass and straightedge construction problems of the ancient Greeks: doubling the cube, trisecting an angle, and squaring the circle. These problems were unsolved by the Greeks. We show that it is not in principle possible to make these constructions, using modern algebraic (and not geometric) techniques. When starting with a line segment of length 1, we call the length of
any line segment we can construct after a finite number of compass and straightedge construction steps to be a constructible number. First, we show that the set of constructible numbers is a field (Corollary 37.3). We can construct all rational numbers (Lemma 37.1 and Theorem 37.2) and all square roots of constructible numbers (Theorem 37.4). This development leads to the Constructible Number Theorem (Theorem 38.3), which asserts that a number α is constructible exactly if the following condition holds: There exists a finite sequence of fields
Q = F0 ⊂ F1 ⊂ · · · ⊂ FN with α ∈ FN and Fi+1 = Fi(
√ ki) for some ki ∈ Fi, with ki > 0 for
i = 0, . . . , N − 1. We show that it is impossible to double the cube by showing that
2 √ 2 is not constructible (Lemmas 39.1 and Theorem 39.2). We show
that it is impossible to trisect a 60◦ angle (that is, construct a 20◦
angle) by showing that to do so would imply being able to construct a solution to x3 − 3x− 1 = 0, which we show is impossible (Lemma 39.3 and Theorem 39.4). Finally, we consider the problem of squaring the circle. If this were possible, then pi would be a constructible number. Lindemann’s Theorem (Theorem 39.5 — which we do not prove) says that pi is transcendental: that is, pi is not the root of any polynomial with rational coefficients. But we show that any constructible number is the root of a polynomial in Q[x] of degree 2n (Theorem 39.7) and so is not transcendental. Thus it is impossible to square the circle.
