| Instructor: Gábor Hetyei | Last update: Wednesday, November 20, 2019 |
| Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercises announced in class and the ones below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise shows up below, I will allow you extra time to hand in the exercise that was announced in class. If, however, exercises are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date. |
Notation: In the table below,
1.1/1a means exercise 1, part a, in section 1.1.
| No. | Date due: | Problems: |
| 13 | Mo Nov 25 |
6.1/2,4,7a,16b. Bonus: Consider the ring of polynomials I with integer coefficients. Let I be the ideal consisting of all polynomials whose constant term is a multiple of k. What is R/I isomorphic to? (B10, 5 points). |
| 12 | Mo Nov 18 |
5.1/10, 12; 5.2/2,8; 5.3/8. Bonus: 5.3/9b (B09, 5 points) |
| 11 | Mo Nov 11 |
4.4/12, 14b (assume r,s and t are pairwise different),
19a; 4.5/1b, 1d; 5.1/4
Bonus:
|
| 10 | Mo Nov 4 |
4.3/8, 22a; 4.4/14a.
Bonus: Suppose R is an integral domain. A non-unit element p in R is prime if whenever p divides a product ab it divides one of the factors. A non-unit element p in R is irreducible if its only divisors are units and associates of p. Prove that every prime is also irreducible. (B07, 5 points) |
9 | Mo Oct 28 | 4.1/5d; 4.2/2,10. |
| 8 | Mo Oct 21 |
3.3/2, 8 (we already know it is a homomorphism, prove only that it is a
bijection); 4.1/12,18,20. Bonus: Let R be a commutative ring with a multiplicative identity element. We say that a is an associate of b in R if there is a unit c satisfying a=bc. Prove that the relation "a is an associate of b" is an equivalence relation. (B06, 5 points) |
| 7 | Mo Oct 14 | 3.3/12a,12b, 24b,26. |
| 6 | We Oct 9 |
3.1/6b, 10, 11bc, 22 (only prove it is a ring);
3.2/3b,20, 22b.
Bonus:
|
| 5 | Mo Sep 30 | 2.3/4a, 4d, 8b (for 8b, use the equations [8][x]=[2], [3][x]=[1] and [6][x]=[4] in ℤ12). |
| 4 | Mo Sep 23 | 2.2/2, 10 (parts 8 and 9), 16a, 16d. |
| 3 | Mo Sep 16 | 2.1/4,10,14a,16. |
| 2 | We Sep 11 | 1.3/6,8b, 14, 16, 19 |
| 1 | We Sep 4 |
1.1/8,11 Board problem: Prove by induction that 12+22+...+n2=n(n+1)(2n+1)/6. 1.2/15c (also write the greatest common divisor of 1003 and 456 as 1003 m+ 456 n, see the file notes-0828.pdf on Canvas for help), 1.2/34a. Bonus Problem: (B01, 5 points) 1.2/33. |