MAT 320 Spring 2008

 You may remember from geometry that you can perform many constructions only using a straightedge and a compass  These include drawing circles, constructing right angles, bisecting angles, etc.  But there are other problems that the ancient Greeks wanted to try to solve with this method

 The Greeks wanted to know if any of the following were possible  Trisecting the angle: Given an angle, divide it into three congruent angles  Doubling the cube: Given a cube, construct another cube with exactly twice the volume  Squaring the circle: Given a circle, create a square with the same area

 It turns out that all of these constructions are impossible  In order to understand why, we need to think about how constructions really work  We start with two points, (0, 0) and (1, 0)  We say that we can “construct” a point (x, y) if we can find that point as an intersection of lines or circles that we can construct

 The things we can construct are  Lines: We can use our straightedge to construct a line between any two points  Circles: Given two points, we can construct a circle with the center at one point and which passes through the other  Perpendiculars: Given a line and a point, we can construct a perpendicular line that passes through the point

 We say that a number is “constructible” if it is the x or y-coordinate of a constructible point  For example, all of the integers are constructible

 The number is also constructible, since the point is the intersection of the first two circles on the previous slide  In fact, the set of constructible numbers is closed under addition, subtraction, multiplication, division, and square roots

 The set of constructible numbers forms a field that contains the rational numbers  This field contains only those numbers that can be obtained from (possibly repeatedly) extending Q with the roots of quadratic polynomials

 For example, Gauss showed that  Since this number is constructed out of rational numbers and square roots, this number must be constructible  We can use this fact to construct a regular 17-sided polygon

 Let’s think about trisection of an angle, specifically a 60-degree angle  60-degree angles are constructible: cos(60) and sin(60) are both constructible numbers  What about 20-degree angles?  Using trig identities, it’s possible to show that cos(20) is a root of the polynomial x 3 – 3x – 1

 Since the polynomial for which cos(20) is a root has degree 3, that means that cos(20) will involve cube roots, which aren’t allowed  So cos(20) is not a constructible number, and 60-degree angles are just one example of angles we cannot trisect with straightedge and compass

 Given a 1 x 1 x 1 cube, we would need to construct a x x cube to have exactly double the volume  But is not a number we can construct, so we wouldn’t be able to create a segment exactly units long to create our cube

 Given a circle of radius 1 (and area π), we would need to construct a square whose sides have length the square root of π  Even though square roots are allowed, π is not a rational number  It turns out π is a transcendental number, which means it’s not the root of any polynomial with rational coefficients

 Another famous impossibility that is related to these ideas is credited to Niels Abel ( )  He proved that there is no way to solve a generic fifth-degree polynomial using radicals (even allowing 5 th roots!)

 Of course, some quintics are solvable using radicals  An example is, whose roots are 1 (twice), -1, i, and –i  But what Abel proved is that there is no analogue to the “quadratic formula” for quintics

 Abel’s proof is beyond what we have learned in this course, but here are some related ideas  Have you ever noticed that roots of polynomials tend to come in groups?  For example, if you know that is the root of a quadratic, you can be sure that is also a root

 It turns out that this is no accident  The roots of higher degree polynomials are related in more complicated ways, but they are still related  Once the degree reaches 5, the relationships become so complicated that there is sometimes no way to “unentangle” the roots from one another

 Keep in mind that we can still solve quintic equations using numerical methods  The issue is that some quintic equations have roots that we cannot express with our normal radical notation  One example is x 5 – x + 1  This does not mean that the roots don’t exist as complex numbers