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 x3 – 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 (1802-1829)

He proved that there is no way to solve a generic fifth-degree polynomial using radicals (even allowing 5th 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 x5 – x + 1

This does not mean that the roots don’t exist as complex numbers