ARCHIVE PAGE

This page is not updated any more.

MATH 3163	MODERN ALGEBRA	Section 002
Instructor: Gábor Hetyei		Last update: December 1, 2000

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.

Date due Problems:

Monday August 28, 2000 A.1/a,d,g, 1.2, 2.1, 2.2, 2.12
Bonus problem: Prove that the relation " A is equivalent to B if there is a 1-1 and ont map from A to B" is an equivalence relation on sets.

Wednesday September 6, 2000 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.18, 3.26, 4.2, 4.8

Monday, September 11 2000 5.6,5.7,5.8,5.9,5.10 and 5.14.

Monday, September 18 2000 6.1/a,b,c,d, 6.2/a,c, 6.4, 7.2, 7.4, 7.13.
Bonus: 6.12 (also due Monday, September 18!), 7.24 (no deadline).

Monday, September 25 2000 Describe the composition of two central projections.
Write up the multiplication table for the symmetries of the square (id,r,r²,r³,t,tr,tr²,tr³; here t denotes the reflection at the vertical axis and r denotes rotation counterclockwise by 90 degree).
8.4, 8.6.
No homework assigned on Friday. Test on Monday!

Monday, October 2 2000 9.2, 9.4, 9.9, 10.7, 10.12

Wednesday, October 11 2000 11.2, 11.4, 12.1, 12.2.

Monday, October 16 2000 13.6 (b), 13.7, 13.8, 14.2, 14.3, 15.2
Bonus: 14.26.

Monday, October 23 2000 15.18, 16.2, 17.2, 17.4, 17.7, 17.10, 17.12, 17.13, 17.17, 17.24
Bonus: 16.8

Monday, October 30 2000 18.2, 18.5/c,d, 18.6/c,d, 18.9, 19.2, 19.5, give an isomorphism between Z₁₀ and the direct product of Z₂ and Z₅. (Z_k stands for the group of integers modulo k.)

Monday November 6, 2000 21.2, 21.6, 21.9, 24.12.
Prove that if a₁ is congruent to a₂ modulo n and b₁ is congruent to b₂ modulo n then a₁b₁ is congruent to a₂b₂ modulo n.
No new hw was assigned on Friday. Test on Monday!

Monday November 13, 2000 25.2, 25.3, 25.4, 26.3, 26.4, 26.13, finish 26.14 (prove that the set given is closed under taking inverse.)
Bonus problem: 25.14.

Monday November 20, 2000 26.2, 26.6, 26.17, 27.14, 27.23, 28.1, 28.7 (c,d,f,g only), 28.6.

Monday December 4, 2000 (No homework was assigned on Monday, November 20.)
29.2, 29.4, 30.5, 30.6, 30.9, 31.7, 31.8, 31.11/b,c, 31.13.
Consider the field of quotients of an integral domain. Prove that if (a₁,b₁ ) is equivalent to (a₂,b₂ ) then (b₁,a₁ ) is equivalent to (b₂,a₂ ). (All elements are assumed to be nonzero.)

Date due	Problems:
Monday August 28, 2000	A.1/a,d,g, 1.2, 2.1, 2.2, 2.12 Bonus problem: Prove that the relation " A is equivalent to B if there is a 1-1 and ont map from A to B" is an equivalence relation on sets.
Wednesday September 6, 2000	3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.18, 3.26, 4.2, 4.8
Monday, September 11 2000	5.6,5.7,5.8,5.9,5.10 and 5.14.
Monday, September 18 2000	6.1/a,b,c,d, 6.2/a,c, 6.4, 7.2, 7.4, 7.13. Bonus: 6.12 (also due Monday, September 18!), 7.24 (no deadline).
Monday, September 25 2000	Describe the composition of two central projections. Write up the multiplication table for the symmetries of the square (id,r,r²,r³,t,tr,tr²,tr³; here t denotes the reflection at the vertical axis and r denotes rotation counterclockwise by 90 degree). 8.4, 8.6. No homework assigned on Friday. Test on Monday!
Monday, October 2 2000	9.2, 9.4, 9.9, 10.7, 10.12
Wednesday, October 11 2000	11.2, 11.4, 12.1, 12.2.
Monday, October 16 2000	13.6 (b), 13.7, 13.8, 14.2, 14.3, 15.2 Bonus: 14.26.
Monday, October 23 2000	15.18, 16.2, 17.2, 17.4, 17.7, 17.10, 17.12, 17.13, 17.17, 17.24 Bonus: 16.8
Monday, October 30 2000	18.2, 18.5/c,d, 18.6/c,d, 18.9, 19.2, 19.5, give an isomorphism between Z₁₀ and the direct product of Z₂ and Z₅. (Z_k stands for the group of integers modulo k.)
Monday November 6, 2000	21.2, 21.6, 21.9, 24.12. Prove that if a₁ is congruent to a₂ modulo n and b₁ is congruent to b₂ modulo n then a₁b₁ is congruent to a₂b₂ modulo n. No new hw was assigned on Friday. Test on Monday!
Monday November 13, 2000	25.2, 25.3, 25.4, 26.3, 26.4, 26.13, finish 26.14 (prove that the set given is closed under taking inverse.) Bonus problem: 25.14.
Monday November 20, 2000	26.2, 26.6, 26.17, 27.14, 27.23, 28.1, 28.7 (c,d,f,g only), 28.6.
Monday December 4, 2000	(No homework was assigned on Monday, November 20.) 29.2, 29.4, 30.5, 30.6, 30.9, 31.7, 31.8, 31.11/b,c, 31.13. Consider the field of quotients of an integral domain. Prove that if (a₁,b₁ ) is equivalent to (a₂,b₂ ) then (b₁,a₁ ) is equivalent to (b₂,a₂ ). (All elements are assumed to be nonzero.)