Algebraic and transcendental numbers. Use the indices in Sethuraman and Courant to find places to browse in those books to supplement the class work.

1. Redo problem 1a from the last homework: show that the formulas

   	a = (u2 - v2)
   	b = 2uv					(PT)
   	c = (u2 + v2)
   

   when u and v are relatively prime integers of opposite parity produce all the primitive Pythagorean triples.

   Your task here is to transcribe your class notes, making sure you convince me that you have convinced yourself (for good reason, not just because I said so) that the logic going from step to step is sound.

2. One of the theorems in Sethuraman that we have not and will not study says that the algebraic numbers form a field. That means that if r and s are numbers and each is a root of some polynomial with integer coefficients (not necessarily the same polynomial!) then both r+s and rs are roots of some other polynomials with integer coefficients. Sethuraman problem 112/3 asks you to find such polynomials for a few algebraic numbers. Some of these may be hard without reading and understanding Sethuraman's proof of the general theorem - which I do not want you to do. Just play around looking for polynomials that will do the job. If you can't manage the hard ones, skip them.)

3. Try Problem 9 on page 114 of Sethuraman. The good news is that this may call for some of the things you learned in calculus. Maybe that's bad news. More bad news is that I think the problem is hard. But give it a try.

4. Prove that there are the same number of integers Z = { ...-3, -2, -1, 0, 1, 2, 3 ...} as positive integers N = {1, 2, 3, ...}. You can use Hilbert's hotel if you like: assign rooms to all the integers to do the matching.

5. Show that there are the same number of polynomials with integer coefficients as there are positive integers. That is, find a one to one matching between Z[x] and N. You can do this by putting up the polynomials as guests in Hilbert's hotel. Be sure to explain the room matching scheme carefully enough so that the night clerk can tell each polynomial where to go.

6. Suppose that each polynomial of degree n comes to the hotel with his family of size n. So, for example, the polynomial f(x) = 7x³ -3x + 423 would need three rooms. Show that Hilbert's hotel can accommodate all the polynomial families.