http://www.math.tamu.edu/~map/courses/627-sp08/

Homework

• HW #1: Due Wed 1/23
  E&W: Ch. 1: 1.1, 1.2, 1.3

• HW #2: Due Fri. 2/1
  E&W: Ch. 1: 1.8, 1.27, 1.28

• HW #3: Due Wed. 2/13
  E&W: Ch. 1: 1.13, 1.15, 1.21, 1.23, 1.24

• HW #4: Due Fri. 2/22
  E&W: Ch. 2: 2.1, 2.2, 2.4, 2.5
  

  (A) Let p be a prime number. For a non-zero rational number x, let ord_p(x) = k, where x = p^k*m/n, for integers m, n both relatively prime to p. We set ord_p(0) = infinity. For x, y rational numebrs, prove the following:

  • ord_p(x*y) = ord_p(x) + ord_p(y);
  • ord_p(x + y) >= min( ord_p(x), ord_p(y) ) always;
  • If ord_p(x) <> ord_p(y) (not equal), then ord_p(x+y) = min( ord_p(x), ord_p(y) ).

  (B) This problem can be done in conjunction with 2.2. Let Z be the integers and Q the rational numbers. Show that the map sending (a,b,c) to (a/c,b/c) from { (a,b,c) in Z^3 | (a,b,c) is a primitive Pythagorean triple } to { (x,y) in Q^2 | x^2 + y^2 = 1 } is a bijection.

• HW #5: Due Fri. 3/7
  E&W: Ch. 2: 2.10
  E&W: Ch. 3: 3.5, 3.8
  

• HW #6: Due Mon. 4/7
  E&W: Ch. 3: 3.9, 3.12, 3.14, 3.15, 3.16

• HW #6: Due Mon. 4/21
  E&W: Ch. 8: 8.6, 8.10, 8.11, 8.16 (can assume s > 2 real) E&W: Ch. 10: 10.2