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MATH 4163	MODERN ALGEBRA	Section 001
Instructor: Gábor Hetyei		Last update: November 22, 2002

Homework assignments

Disclaimer: The information below comes with no warranty. If, due to typocraphical error, there is a discrepancy between the exercise numbers announced in class and the numbers 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 number shows up below, I will allow you extra time to hand in the exercise whose number was announced in class. If, however, exercise numbers are missing because this page is not up to date, it is your responsability 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.

No. Date due Problems
14 Monday November 25, 2002 5.1/2,4ab,8,9cd, 5.3/6bce.

13 Monday November 18, 2002 3.10/3, 3.11/8,9. 5.1/1. Bonus: Using Gauss' lemma, prove that the square root of a positive integer is either an integer, or it is irrational.

12 Monday November 11, 2002 3.7/3, 3.9/1a. Bonus: 3.9/6.

11 Monday November 4, 2002 3.8/1,2,4,5. Prove that d(ab)=d(a)d(b) for Gaussian integers and the norm defined in the book. Bonus: Prove that if the Gaussian integer a+bi is a prime then a²+b² is a prime integer.

10 Monday October 28, 2002 3.5/3, 3.6/3,6, 3.7/6,7,8.

9 Monday October 21, 2002 3.2/6 (assume nonzero characteristic), 3.4/5,6,7, 3.5/1. Bonus: Prove that the ring of n×n matrices is simple.

8 Monday October 14, 2002 2.12/11a, 2.13/5, 2.14/1,4cd, 3.2/4. Bonus: 2.13/9.

7 Monday October 7, 2002 2.12/2, 2.13/1,4a. List the 3-Sylows of the symmetric group on 4 elements. Prove that the only group with p (prime) number of elements is the cyclic group of order p. Determine the 2-Sylow subgroups of the dihedral groups D₄ and D₆. Prove that the direct product of the cyclic groups of order 3 and order 5 is a cyclic group of order 15. When is the direct product of the cyclic groups of order m and n isomorphic to a cyclic group of order mn ?

6 Monday September 30, 2002 2.11/2,4 (for n=2 and n=3 only), supplementary problem 20a from the end of Chapter 2 (p. 117). Prove that if p divides n then the cyclic group of order n contains an element of order p. Bonus: Supplementary problems 21,22,23 from the end of Chapter 2. (Expires Monday September 30, 2002.)

5 Monday September 23, 2002 2.9/5,7, 2.10/1b,2,3,6bc,9,10. Bonus: 2.9/6, prove that (a₁,a₂,...,a_n) =(a₁,a₂)(a₁,a₃)... (a₁,a_n), write the transposition (i,j)) as a product of transpositions of the form (k,k+1).

4 Monday September 16, 2002 2.7/5a, 15, 2.8/1cd,8. Bonus: 2.7/17.

3 Monday September 9, 2002 2.6/9,17, 2.7/1cde,3,5.

2 Wednesday September 4, 2002 2.5/2,5 (use 4), 15, 16, 2.6/2, 3. Bonus: 2.5/9. Note: 2.5/14 was assigned earlier this week, but we ended up doing it together in class.

1 Monday August 26, 2002 1.1/1b (p. 8), 2.3/1bc,7,10,15 (p. 35-36). Bonus: Prove that the left zero semigroup is a semigroup, that is, the multiplication xy=x is associative; 2.3/14 (p. 36).

No.	Date due	Problems
14	Monday November 25, 2002	5.1/2,4ab,8,9cd, 5.3/6bce.
13	Monday November 18, 2002	3.10/3, 3.11/8,9. 5.1/1. Bonus: Using Gauss' lemma, prove that the square root of a positive integer is either an integer, or it is irrational.
12	Monday November 11, 2002	3.7/3, 3.9/1a. Bonus: 3.9/6.
11	Monday November 4, 2002	3.8/1,2,4,5. Prove that d(ab)=d(a)d(b) for Gaussian integers and the norm defined in the book. Bonus: Prove that if the Gaussian integer a+bi is a prime then a²+b² is a prime integer.
10	Monday October 28, 2002	3.5/3, 3.6/3,6, 3.7/6,7,8.
9	Monday October 21, 2002	3.2/6 (assume nonzero characteristic), 3.4/5,6,7, 3.5/1. Bonus: Prove that the ring of n×n matrices is simple.
8	Monday October 14, 2002	2.12/11a, 2.13/5, 2.14/1,4cd, 3.2/4. Bonus: 2.13/9.
7	Monday October 7, 2002	2.12/2, 2.13/1,4a. List the 3-Sylows of the symmetric group on 4 elements. Prove that the only group with p (prime) number of elements is the cyclic group of order p. Determine the 2-Sylow subgroups of the dihedral groups D₄ and D₆. Prove that the direct product of the cyclic groups of order 3 and order 5 is a cyclic group of order 15. When is the direct product of the cyclic groups of order m and n isomorphic to a cyclic group of order mn ?
6	Monday September 30, 2002	2.11/2,4 (for n=2 and n=3 only), supplementary problem 20a from the end of Chapter 2 (p. 117). Prove that if p divides n then the cyclic group of order n contains an element of order p. Bonus: Supplementary problems 21,22,23 from the end of Chapter 2. (Expires Monday September 30, 2002.)
5	Monday September 23, 2002	2.9/5,7, 2.10/1b,2,3,6bc,9,10. Bonus: 2.9/6, prove that (a₁,a₂,...,a_n) =(a₁,a₂)(a₁,a₃)... (a₁,a_n), write the transposition (i,j)) as a product of transpositions of the form (k,k+1).
4	Monday September 16, 2002	2.7/5a, 15, 2.8/1cd,8. Bonus: 2.7/17.
3	Monday September 9, 2002	2.6/9,17, 2.7/1cde,3,5.
2	Wednesday September 4, 2002	2.5/2,5 (use 4), 15, 16, 2.6/2, 3. Bonus: 2.5/9. Note: 2.5/14 was assigned earlier this week, but we ended up doing it together in class.
1	Monday August 26, 2002	1.1/1b (p. 8), 2.3/1bc,7,10,15 (p. 35-36). Bonus: Prove that the left zero semigroup is a semigroup, that is, the multiplication xy=x is associative; 2.3/14 (p. 36).