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Homework assignments
(MATH 3163-002, Spring 2022)
Instructor: Gábor Hetyei Last update: Thursday, April 21, 2022

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.

The deadlines do not apply to the Bonus questions, which expire only once we solve them in class, or on April 21 at latest.

Notation: In the table below, 1.1/1a means exercise 1, part a, in section 1.1.

No. Date due: Problems:
13 Tue Apr 26 at noon 6.1/7ab, 8, 16b.
12 Tue Apr 19 at noon 5.3/8, 10;   6.1/4, 6a.
Bonus problem:
  1. 5.3/9b (B09, 5 points)
11 Tue Apr 12 at noon 4.5/1b, 1d   5.1/4,10,12.
10 Tue Apr 5 at noon 4.6/2
9 Tue Mar 29 at noon 4.4/12,14a (you may assume that r and s are different),19a (you may use the product rule for derivatives),24.
8 Tue Mar 22 at noon 4.2/2,8,10;   4.3/6,10a.
Bonus problem:
  1. Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab). (B08, 10 points)
7 Tue Mar 15 at noon 3.3/8,30   4.1/5d,12,18,20
Bonus problem:
  1. 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. (B07, 5 points)
6 Tue Mar 1 at noon 3.2/3b,20, 22b;   3.3/12ab, 24b, 26.
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. (B06, 10 points).
5 Tue Feb 22 at noon 3.1/6b,10, 11bc,22 (only prove it is a ring, distributive law on one side suffices).
Bonus:
  1. 2.3/13b (B04, 5 points)
  2. 2.3/14a (B05, 5 points)
4 Tue Feb 15 at noon 2.3/2,4,10.
Bonus:
  1. (B03, 3 points) Prove that any nonzero [a] in ℤn is either a unit or a zerodivisor.
3 Tue Feb 8 at noon 2.1/16   2.2/2,4,10,14c.
2 Tue Feb 1 at noon 1.3/6, 8b, 14;   2.1/4, 5b, 10.
Bonus:
  1. Prove that the square root of a positive integer is either an integer or it is irrational. (B02, 5 points)
1 Tue Jan 25 at noon 1.1/8,11;   1.2/15c (also write the greatest common divisor of 1003 and 456 as 1003 m+ 456 n, see the file notes-0120.pdf on Canvas for help).

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.
Bonus:
  1. 1.2/33. (B01, 5 points)