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Homework assignments
(MATH 3163-001, Spring 2021)
Instructor: Gábor Hetyei Last update: Tuesday, April 22, 2021

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 Thu 4/29 at noon 6.1/4, 6a, 7a, 16b.
11 Thu 4/22 at noon 5.2/2,8;   5.3/8,10.
Bonus: 5.3/9b (B11, 10 points)
10 Thu 4/15 at noon 4.4/19a;   4.6/2a, 2b;   5.1/4,10,12.
9 Thu 4/8 at noon 4.4/12, 14a, 24;   4.5/1b, 1d.
8 Thu 4/1 at noon 4.3/6,10a,22a.
7 Thu 3/25 at noon 4.1/5d,12,18,20;   4.2/2,8,10.
Bonus:
  1. 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. Prove you have an isomorphism. (B08, 10 points)
  2. 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. (B09, 5 points)
  3. Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab). (B10, 7 points)
6 Thu 3/18 at noon 3.3/2,8,12a, 12b, 24b, 26.
Bonus: (B07, 5 points) Prove that for any ring element a, and any pair of positive integers m,n, we have m(na)=(mn)a.
5 Thu 3/11 at noon 3.1/6b,10,11bc,22;   3.2/3b, 20, 22b.
Bonus: (B06, 3 points) Prove that for any nonzero [a] in ℤn, if a and n are not relative primes then [a] is a zerodivisor.
4 Thu 3/04 at noon 2.2/2, 10 (parts 8 and 9), 16a, 16d;   2.3/4a, 4d, 8b (for 8b use the equations [8][x]=[2], [3][x]=[1] and [6][x]=[4] in ℤ12).
3 Thu 2/25 at noon 2.1/5b, 16;   2.2/4,2,14c.
Bonus:
  1. 1.2/33. (B01, 5 points)
  2. Prove that the square root of a positive integer is either an integer or it is irrational. (B02, 5 points)
  3. 2.1/20 for n=6 (B03, 5 points)
  4. 2.3/13b (B04, 5 points)
  5. 2.3/14a (B05, 5 points)
2 Thu 2/18 at noon 1.2/4a, 15c (also write the greatest common divisor of 1003 and 456 as 1003 m+ 456 n, see the file notes-0202.pdf on Canvas for help), 34a;   1.3/6, 8b, 14;   2.1/4, 10.
1 Tu 2/2 at noon 1.1/8,11.
Board problems:
  1. Prove by induction that 12+22+...+n2=n(n+1)(2n+1)/6.
  2. The sequence a0, a1, a2,... is given by the initial conditions a0=4 , a1=7 and by the recurrence an+2=3 · an+1-2 ·an.
    Prove by induction that an=1+3 · 2n.
  3. Prove that a divides b if and only if a divides -b