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Homework assignments
(MATH 3163-001, Spring 2015)
Instructor: Gábor Hetyei Last update: Monday, April 13, 2015

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 4/22 6.1/2, 4, 7a, 16b.
12 4/15 5.2/2,6,8,10;   5.3/8
Bonus problems:
  1. (B9) Use the Euclidean algorithm to find the multiplicative inverse of the class of 5x+1 in the ring of polynomials with rational coefficients modulo x2-2. (3 points)
  2. (B10) 5.3/9b (3 points)
11 4/8 4.4/4a, 4b   4.5/1b, 1d;   5.1/2,4,6,10.
10 4/1 4.4/12, 14a, 14b, 19a.
Bonus questions:
  1. (B5) 4.4/24 (5points)
  2. (B6) Tell which earlier homework exercise is a special case of 4.4/24 (2 points).
  3. (B7) Find the theorems in the book that imply that, in the ring of polynomials with real coefficients, every irreducible polynomial has degree at most 2. Also find the proof of the fact that the complex conjugate of a root of a polynomial with real coefficients is a root of the same polynomial (3 points).
  4. (B8) Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab, 6 points).
9 3/25 4.3/2,4,6,8.
8 3/18 4.1/12, 18, 20;   4.2/2,10.
Board problem: Let F be a field. Prove that the relation " f(x) is an associate of g(x)" is an equivalence relation on the set F[x]. (The definition of associates is at the beginning of section 4.2.)
7 3/11 3.3/2, 12a, 12b, 24b, 26;   3.1/28, 39.
Bonus: (B4) Find an isomorphism between the ring defined in 3.1/22 and the ring of integers with the usual addition and multiplication. Prove that your bijection is an isomorphism. (5 points)
6 2/25 3.2/3b, 20, 22a, 26.
5 2/23 3.1/8, 12, 22, 36, 40.
4 2/11 2.2/2, 16a, 16d;   2.3/4a, 4d, 8b.
Bonus problems:
  1. (B2) 2.3/13b (5 points)
  2. (B3) 2.3/14a (5 points)

Our first test is on Monday February 9. Please have a look at the Sample Test 1 when you prepare for the test.
3 2/4 1.3/22,26,27,31;   2.1/4,7.
Bonus problem: (B1) Prove that the square root of any positive integer is either an integer or an irrational number. (5 points)
Note: In class I also assigned 1.3/31, but forgot to put it on this list until Wednesday, February 4, 2015. If you did not do 1.3/31 please turn it in by Wednesday, February 11, with your next assignment.
2 1/28 1.2/15c, 34a;   1.3/6,14.
1 1/21 1.1/8,11;   1.2/11, 16.
Board problem: Express 4 as an integer linear combination of 148 and 524. (See Exercise 1.2/15 and your notes from class).