Since a != 0 and Z_prime is a field, a has a multiplicative inverse c. Then substitute x = -bc to see that we've found one root. To prove that there can't be more than one, suppose x and y are both roots. We'll prove that x=y:

	ax + b = 0 = ay + b

	ax = ay

	c(ax) = c(ay)  ( where c is the multiplicative inverse of a )

	x = 1x = (ca)x = c(ax) = c(ay) = ... = y

This argument clearly works in Z_n even when n is composite for those values of a that happen to have a multiplicative inverse. We know those are the ones for which gcd(a,n)=1. So for example in Z₁₀[x] the linear polynomials x+b, 3x+b, 7x+b and 9x+b all have just one root for any of the 10 possible values for b. But we saw in class that the polynomial 5x has more than one root, because both 0 and 2 are roots. In fact, in this case that polynomial has five roots: 0, 2, 4, 6 and 8!

We wondered whether in the composite case the average number of roots would turn out to be 1, because even though some linear polynomials had too many roots, others could have none. Lewanda wanted that as a homework problem, so here it is. See how far you can get with it. You might start with the 10 polynomials 5x+b in Z₁₀[x]. Try to discover when those polynomials have any roots at all, and, when they do, how many they have. You might want to look at some examples in Z₁₂[x] too when looking for patterns.

You may skip the starred item 3 on page 125 on Apollonius's Problem.

Courant and Robbins sprinkle many exercises along the way. They are mostly just computations to make concrete some of the more abstract text. I think most of them are worth working on, but they might involve too much tedious algebra and arithmetic. If you find yourself spending time on boring stuff that isn't increasing your understanding of the text, and you do really understand the text, you can say so in your homework and move on to the next exercises.

1. Do the exercise on page 125 if you can, but don't spend a lot of time on it if you can't do it quickly.

2. Do the exercise at the top of page 128. It should be just a few lines of argument - not much more than the hint there already.

3. Work out the exercise(s) near the bottom of page 128. This one's just arithmetic - you could assign it to your students.

4. Describe at least one of the constructions asked for in the exercise in the middle of page 129. THe description should tell me how to do the job, in a form like that Courant and Robbins use on page 121.

5. Skip the exercises at the top of page 130 (if you want to). Do the first of the two exercises near the top of page 132.

6. Prove that the side of the 2^m-gon and at least one of the other numbers specified in the exercises on page 133 are constructible. Use as your model prose the paragraph beginning "The following example may illustrate the process ..." at the top of that page.