Ideals and Congruence
The most relevant part of the current definitions and theorems file, is pages 9 and 10
Friday, June 5: Ideals, Principal Ideals, Cosets and Congruence
Videos lectures:
✎ Assignment:
1. I have
proven or mosly proven all of the theorems in 64-74 in these
videos. Choose three theorems from three different videos to
write up and turn in for this assignment.
6.1 # 1, 4, 11a, b, c, 23 a, b*, 32a**
* for #23, do not do all of the steps to prove that each set I is an
ideal. Each of these is a principal ideal, so all you need to do
is to write it in the format of theorem 65 to show that it is an ideal.
**OK to explain 32a with examples.
Monday, June 8: Ideals and Congruence in
Polynomial Rings. Note: Meeting time moves to 9:00 am starting today
Video lectures:
✎ Assignment:
5.1 # 1 a,c, 4, 5, 6
5.2 # 5, 7
Tuesday, June 9, study for the quiz.
The quiz will include questions about:
- Given a subset of a ring, prove whether it is or is not a subring, and is or is not an ideal
- List all of the distinct principal ideals of a Zn ring
- List all of the distinct cosets of an ideal in a Zn ring
- Show that two polynomials are or are not congruent mod some principal ideal in a polynomial ring (see 5.1 problem 1)
- Describe the congruence classes in a polynomial ring mod some principal ideal
- Describe the addition and multiplication rules for a polynomial ring mod some principal ideal
- Answer questions about one of these theorems or their proofs:
- Theorem 66 or 67
- Theorem 72 or 73
Wenesday, June 10, take the quiz