| Instructor: Gábor Hetyei | Last update: Tuesday, April 12, 2016 |
| 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: |
| 12 | 4/19 | 6.1/2, 4, 7a, 16b. |
| 11 | 4/12 |
5.2/2,6,8,10; 5.3/8 Bonus:
|
| 10 | 4/5 | 5.1/2,4,6,10 |
| 9 | 3/29 |
4.3/22a, 23b; 4.5/1b, 1d. Our second test is on Tuesday, March 29. You may download the Sample Test II to prepare. |
| 8 | 3/22 |
4.3/2,4,6,8; 4.4/12, 14a, 14b, 19a.
Bonus:
|
| 7 | 3/15 |
4.1/12,18,20; 4.2/2,10 Bonus: (B7) Let F be a field, we say that a polynomial f(x) in F[x] is an associate of g(x) in F[x] if there is a nonzero constant c satisfying f(x)=cg(x). Prove that the relation "f(x) is an associate of g(x)" is an equivalence relation. |
| 6 | 3/1 | 3.3/2, 12a, 12b, 24b, 26. |
| 5 | 2/23 |
3.2/3b, 20, 22a, 26.
Bonus: (B6) Prove that the square root of a positive integer is either an integer or it is irrational. |
| 4 | 2/16 |
3.1/6b, 10, 11b, 11c, 22 (only prove it is a ring, no need to show it is
an integral domain). Bonus: (B5) Find an isomorphism (=a bijection that is compatible with addition and multiplication) between the ring of integers (with the usual addition and multiplication) and the ring in exercise 3.1/22. Our first test is on Tuesday, February 16. You may download the Sample Test I to prepare. |
| 3 | 2/9 |
2.3/4a, 4d, 8b (for 8b, use the
equations [8][x]=[2], [3][x]=[1]
and [6][x]=[4] in ℤ12);
3.1/8,12. Bonus: 2.1/20 with n=6 (B2), 2.3/13b (B3), 2.3/14a (B4) |
| 2 | 2/2 | 2.1/4,7 2.2/2, 10 (prove parts 8 and 9 only), 16a, 16d. |
| 1 | 1/26 |
1.1/8,11; 1.2/15ac, 34a. In 1.2/15 also express the greatest
common divisor as an integer linear combination of the original pair of
numbers. 1.3/6,14. Board problem: Prove by induction that 12+22+...+n2=n(n+1)(2n+1)/6. Bonus problem: (B1) 1.2/33. (You are allowed to use the hints in the back, but not allowed to use the Fundamental Theorem of Arithmetic from the next section.) |