MAS 6396, CONSTRUCTIVE ALGEBRA
- 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.
- 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
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
- 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
- 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