Geometric Constructions

In ancient Greece, three classic problems were posed. These problems are geometric in nature and involve straightedge-and-compass constructions from what is now high school geometry; that is, we are allowed to use only a straightedge and compass to solve them. The problems can be stated as follows.

1. Given an arbitrary angle, can one trisect the angle into three equal subangles using only a straightedge and compass?

2. Given an arbitrary circle, can one construct a square with the same area using only a straightedge and compass?

3. Given a cube, can one construct the edge of another cube having twice the volume of the original? Again, we are only allowed to use a straightedge and compass to do the construction.

First, let us determine more specifically what we mean by a straightedge and compass, and also examine the nature of these problems in a bit more depth. To begin with, a straightedge is not a ruler. We cannot measure arbitrary lengths with a straightedge. It is merely a tool for drawing a line through two points. The statement that the trisection of an arbitrary angle is impossible means that there is at least one angle that is impossible to trisect with a straightedge-and-compass construction. Certainly it is possible to trisect an angle in special cases. We can construct a \(30^\circ\) angle; hence, it is possible to trisect a \(90^\circ\) angle. However, we will show that it is impossible to construct a \(20^\circ\) angle. Therefore, we cannot trisect a \(60^\circ\) angle.

A real number \(\alpha\) is constructible if we can construct a line segment of length \(| \alpha |\) in a finite number of steps from a segment of unit length by using a straightedge and compass.

By , we can locate in the plane any point \(P =( p, q)\) that has rational coordinates \(p\) and \(q\). We need to know what other points can be constructed with a compass and straightedge from points with rational coordinates.

Considering the case of the intersection of a line and a circle, we must determine the nature of the solutions of the equations \[\begin{aligned}a x + by + c & = 0 \\ x^2 + y^2 + d x + e y + f & = 0\end{aligned}\]. If we eliminate \(y\) from these equations, we obtain an equation of the form \(Ax^2 + B x + C = 0\), where \(A\), \(B\), and \(C\) are in \(F\). The \(x\) coordinate of the intersection points is given by \[\begin{aligned}\end{aligned}\] and is in \(F( \sqrt{\alpha}\, )\), where \(\alpha = B^2 - 4 A C \gt 0\). We have proven the following lemma.

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

We are now ready to investigate the classical problems of doubling the cube and squaring the circle. We can use the field of constructible numbers to show exactly when a particular geometric construction can be accomplished.

Given the edge of the cube, it is impossible to construct with a straightedge and compass the edge of the cube that has twice the volume of the original cube. Let the original cube have an edge of length \(1\) and, therefore, a volume of \(1\). If we could construct a cube having a volume of \(2\), then this new cube would have an edge of length \(\sqrt[3]{2}\). However, \(\sqrt[3]{2}\) is a zero of the irreducible polynomial \(x^3 -2\) over \({\mathbb Q}\); hence, \[\begin{aligned}\end{aligned}\] This is impossible, since \(3\) is not a power of \(2\).

Suppose that we have a circle of radius \(1\). The area of the circle is \(\pi\); therefore, we must be able to construct a square with side \(\sqrt{\pi}\). This is impossible since \(\pi\) and consequently \(\sqrt{\pi}\) are both transcendental. Therefore, using a straightedge and compass, it is not possible to construct a square with the same area as the circle.

Trisecting an arbitrary angle is impossible. We will show that it is impossible to construct a \(20^\circ\) angle. Consequently, a \(60^{\circ}\) angle cannot be trisected. We first need to calculate the triple-angle formula for the cosine: \[\begin{aligned}\cos 3 \theta & = \cos( 2 \theta + \theta ) \\ & = \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta \\ & = ( 2 \cos^2 \theta - 1) \cos \theta - 2 \sin^2 \theta \cos \theta \\ & = ( 2 \cos^2 \theta - 1) \cos \theta - 2 (1- \cos^2 \theta) \cos \theta \\ & = 4 \cos^3 \theta - 3 \cos \theta\end{aligned}\]. The angle \(\theta\) can be constructed if and only if \(\alpha = \cos \theta\) is constructible. Let \(\theta = 20^{\circ}\). Then \(\cos 3 \theta = \cos 60^\circ = 1/2\). By the triple-angle formula for the cosine, \[\begin{aligned}\end{aligned}\]. Therefore, \(\alpha\) is a zero of \(8 x^3 - 6 x -1\). This polynomial has no factors in \({\mathbb Z}[x]\), and hence is irreducible over \({\mathbb Q}[x]\). Thus, \([{\mathbb Q}( \alpha ) : {\mathbb Q }] = 3\). Consequently, \(\alpha\) cannot be a constructible number.

Extensions of the field of rational numbers are a central object of study in number theory, so with Sage's roots in this discipline, it is no surprise that there is extensive support for fields and for extensions of the rationals. Sage also contains an implementation of the entire field of algebraic numbers, with exact representations.

Algebraic number theory uses the tools of algebra to solve problems in number theory. Modern algebraic number theory began with Pierre de Fermat (16011665). Certainly we can find many positive integers that satisfy the equation \(x^2 + y^2 = z^2\); Fermat conjectured that the equation \(x^n + y^n = z^n\) has no positive integer solutions for \(n \geq 3\). He stated in the margin of his copy of the Latin translation of Diophantus' Arithmetica that he had found a marvelous proof of this theorem, but that the margin of the book was too narrow to contain it. Building on work of other mathematicians, it was Andrew Wiles who finally succeeded in proving Fermat's Last Theorem in the 1990s. Wiles's achievement was reported on the front page of the New York Times.

Attempts to prove Fermat's Last Theorem have led to important contributions to algebraic number theory by such notable mathematicians as Leonhard Euler (17071783). Significant advances in the understanding of Fermat's Last Theorem were made by Ernst Kummer (18101893). Kummer's student, Leopold Kronecker (18231891), became one of the leading algebraists of the nineteenth century. Kronecker's theory of ideals and his study of algebraic number theory added much to the understanding of fields.

David Hilbert (18621943) and Hermann Minkowski (18641909) were among the mathematicians who led the way in this subject at the beginning of the twentieth century. Hilbert and Minkowski were both mathematicians at Göttingen University in Germany. Göttingen was truly one the most important centers of mathematical research during the last two centuries. The large number of exceptional mathematicians who studied there included Gauss, Dirichlet, Riemann, Dedekind, Noether, and Weyl.

André Weil answered questions in number theory using algebraic geometry, a field of mathematics that studies geometry by studying commutative rings. From about 1955 to 1970, Alexander Grothendieck dominated the field of algebraic geometry. Pierre Deligne, a student of Grothendieck, solved several of Weil's number-theoretic conjectures. One of the most recent contributions to algebra and number theory is Gerd Faltings' proof of the Mordell conjecture. This conjecture of Mordell, now known as Faltings' theorem, essentially says that certain polynomials \(p(x, y)\) in \({\mathbb Z}[x,y]\) have only a finite number of integral solutions.