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Homework assignments
(MATH 3163-001, Fall 2022)
Instructor: Gábor Hetyei Last update: Tuesday, November 22, 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 Nov 29, 10:00 am 5.3/8,10.
Bonus problem:
  1. 5.3/9b (B12, 7 points)
12 Tue Nov 22, 10:00 am 5.1/4,10,12;   5.2/2. In 5.2/2, multiplication table suffices, but it should be in simplest form: all entries should be of the form ax+b, there should be no entry containing x2.
11 Tue Nov 15, 10:00 am 4.5/1b,1d,2;   4.6/2.
Bonus problem:
  1. Prove the product rule for derivatives for polynomials with coefficients in an arbitrary field. (11.5/5ab). (B11, 10 points)
10 Tue Nov 8, 10:00 am 4.2/10;   4.4/12,14a (you may assume that r and s are different),19a (you may use the product rule for derivatives).
Bonus problem:
  1. Let R be an integral domain. We say that p is irreducible, if its only divisors are the units and the assoicates of p (that is, elements of the form up where u is a unit. We say that p is a prime if whenever p is a factor of ab, it is also a factor of a or b. Prove that every prime is irreducible. (B10, 5 points)
9 Th Nov 3, 10:00 am 4.3/6, 10a, 12.
8 Tue Oct 25, 10:00 am 4.1/5d, 12, 18, 20;   4.2/2,8;   4.3/2.
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. (B09, 5 points)
7 Tue Oct 18, 10:00 am 3.3/8,30.
Bonus problem:
  1. Prove that m(na)=(mn)a holds in any ring, for any ring element a and any pair of integers m, n. (B08, 5 points)
6 Th Oct 13, 10:00 am 2.3/2,4;   3.3/12a, 12b, 24b, 26.
Bonus problems:
  1. Prove that any nonzero [a] in ℤn is either a unit or a zerodivisor. (B06, 3 points)
  2. Prove Napoleon's Theorem. (See the handout Napoleon's theorem. B07, 10 points)
5 Tue Oct 4, 10:00 am 3.2/3b,20,22b and the following Board problem:
Assume a is an idempotent element (see 3.2/3 for the definition of an idempotent element). Prove by induction on n that an=a holds for n≥ 1.
Bonus problems:
  1. Prove that the square root of a positive integer is either an integer or it is irrational. (B03, 5 points)
  2. 2.3/13b (B04, 5 points)
  3. 2.3/14a (B05, 5 points)
4 Th Sep 29, 10:00 am 3.1/6b,10, 11bc,22 (only prove it is a ring, distributive law on one side suffices).
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. (B02, 10 points).
3 Tue Sep 20, 10:00 am 2.1/16   2.2/2,4,10,14c.
2 Tue Sep 13, 10:00 am 1.3/6, 8b, 14;   2.1/4, 5b, 10.
Bonus: 1.2/33, without using unique prime factorization (B01, 5 points).
1 Tue Sep 6, 10:00 am 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-0901.pdf on Canvas for help);   and the following
Board problems:
  1. Prove by induction that 12+22+...+n2=n(n+1)(2n+1)/6. holds for n ≥ 1.
  2. The sequence a0, a1, a2,... is given by the initial conditions a0=4 , a1=-1 and by the recurrence an+2= an+1+2 ·an.
    Prove by induction that an=2n+3 · (-1)n.
  3. Prove that b divides a if and only if -b divides a.