MAS 6396, CONSTRUCTIVE ALGEBRA

Fall 2012

Assignments

• Page 6, Exercises 1 through 3. Do these to hand in on Monday, August 27.
• Try these to hand in on Wednesday, September 5. I picked relatively simple algebraic exercises dealing with Brouwerian examples.
  • Page 6, Exercises 4 and 5. In Exercise 5, the statement "neither of the original sequences does" means that you can't prove that either of the original sequences does.
  • Page 41, Exercise 9.
  • Page 46, Exercise 2.
• Click here for a solution to Exercise I.1.5.
• For Monday, September 10.
  • Page 6, Exercises 6 and 8. For Exercise 8, try a tree with a single branch point.
  • Page 48, Exercise 14.
• For Monday, September 17.
  • Page 48, Exercise 17.
• Click here for a solution to Exercise II.2.17.
• For Monday, September 24.
  • Page 48, Exercise 15 (make that a "finitely generated left ideal").
  • Read Section II.3 on the real numbers. Do Exercises 1 and 2 on page 51.
  • Click here for a partial solution to Exercise II.3.2.
• For Monday, October 1.
  • Page 51, Exercise 5. Just show that each of (ii) through (v) implies LLPO.
• For Monday, October 8.
  • Page 79, Exercises 1 and 2.
• For Monday, October 15.
  • Page 79, Exercises 3, 4, 5, and 6.
• For Monday, October 22
  • Page 69, Exercises 1 and 4.
• For Monday, October 29
  • Page 72, Exercises 1 and 3.
• A web search turned up this note about the Cayley-Hamilton theorem and polynomials over a noncommutative ring. The fact that it is from NMSU, where I taught for many years, is purely coincidental.
• For Monday, November 5
  • Page 80, Exercises 8 and 9.
  • Page 84, Exercise 2.
• For Wednesday, November 14
  • Redo the last assignment. Make your statements intelligible. That's more important than getting them right---we can work on that later.
  • Make it clear whether you are claiming something for some r, or whether you are claiming it for all r, or whether you are claiming it for the r you introduced a couple of lines previously.
  • Don't use the terminology "principally generated ideal" with me. Other people might be fond of it, but I can't stand it. Those things ] are almost universally called "principal ideals". There is some argument for calling them "1-generated ideals", but even that is pointless unless perhaps you are in context where you are also talking about 2-generated ideals and 3-generated ideals.
  • Learn what IM means when I is an ideal in a (commutative) ring R and M is an R-module. In particular, learn what the product IJ of two ideals in R is.
  • Learn why a finitely generated ideal is principal both in the integers, and in the polynomial ring k[X] over a discrete field k. Hint: It is not because every ideal in those rings is principal.
• For Wednesday, November 21
  • Page 84, Exercises 3, 4, and 5. Note: For Exercise 5, the definition of I:S is given on page 44. It's called a quotient.
• For Wednesday, November 28
  • Page 85, Exercise 11.