From Euclid to Spigotry
Algebra and Constructions
Last modified: September 10, 2004
The ancient Greeks were fascinated by the problem of what can be constructed with only a straightedge and a compass.  Later developments in algebra showed that this question is closely related to the concept of a "field."  
References:
Beckmann, Chapter 4 and pp. 166-167
Bold, Chapters 1 and 2
Lecture topic 1: Fields
Define a field (informally, along the lines of Bold Chapter 2). Explain the connection between the requirements for a field (Bold, page 8) and what you have been taught about how to do arithmetic with fractions.
Lecture topic 2: Square roots are irrational
Prove that  is not a rational number (see p. 167 of Beckmann).  Then extend the proof to
 for any positive integer k that is not a perfect square.
Lecture topic 3: The algebra of intersecting lines
a. Suppose that you start with two points whose coordinates are rational numbers. Show that the line joining the two points can be expressed in terms of rational numbers.
b.  Show that the problem of finding the intersection of two such lines, regarded as a problem in analytic geometry, leads to points whose coordinates are rational numbers. 

Lecture topic 4: The algebra of intersecting lines and circles
a. Show that the problem of finding the intersection of a line and a circle requires the extraction of at most one square root.  Without loss of generality, you may assume that the circle has its center at the origin and that the line is horizontal.
b. Show that the problem of finding the intersection of two circles requires the extraction of at most one square root.  Without loss of generality, you may assume that one circle has its center at the origin and that the other has its center on the y-axis.
Lecture topic 5: A simple field extension
a. Show how to express  in the form x = a + b by "rationalizing the denominator."
b. Show that if a number is of the form x = a + b (where a and b are rational but  is not), then 1/x is also of the same form. Remember: you can't divide by zero!

Lecture topic 6: A more complicated field extension
a. Show how to construct a segment of length  using compass and straightedge.
b. Show that x =  is a solution to the equation  (Hint: twice isolate a square root on the right hand side, then square.
c. Suppose that you carry out a construction in which you need to determine the intersection of a circle and a straight line twice. What general form might the coordinates of a point now take?
Lecture topic 7: Constructible numbers
a. Suppose that you carry out a complicated construction in which you need to determine the intersection of a circle and a straight line three times. What form might the coordinates of a point take?
b. State the general rule (Bold, pp. 12 and 13) for constructible numbers.
c. At the age of 19, Carl Friedrich Gauss discovered that the cosine of one of the angles in a regular heptadecagon (17-sided polygon) is given by the formula
.
What does this tell you about the constructibility of the regular heptadecagon?