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MATH 3163	MODERN ALGEBRA	Section 001
Instructor: Gábor Hetyei		Last update: April 26, 2001

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:

Tuesday January 23, 2001 A.1/a,d,g,h,i, 1.2, 1.4, 1.24.

Tuesday January 30, 2001 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 4.2
Bonus: 3.20, Prove that the relation " A is equivalent to B if there is a 1-1 and onto map from A to B" is an equivalence relation on sets.

Tuesday February 6, 2001 4.8, 5.6, 5.7, 5.8, 6.1/a,b,c,d, 6.2/a,c, 6.4, 7.2.
Bonus: Prove that a map from S to S has a left inverse if and only if it is 1-1, and it has a right inverse if and only if it is onto.

Tuesday February 15, 2001 7.4, 7.2 (again!), 8.2, describe the composition of two central reflections.
Bonus: 7.24.

Tuesday February 27, 2001 9.2, 9.9, 10.1, 10.12.
Bonus: 10.16. Solve the Gear Puzzle I cited from the computergame Myst in class all show that it has no solution.

Tuesday March 13, 2001 11.2, 11.4, 11.6, 11.8, 12.1, 12.2.
Bonus: Let r₁, r₂, r₃, and r₄ be the first four remainders in the Euclidean algorithm. Express them as a linear combination of a and b.

Tuesday March 20, 2001 13.2, 13.4, 13.6 (b), 13.11 (c) (draw also a picture indicating which divisor divides which other, as seen in class), 14.2, 14.4, 14.10.

Tuesday March 27, 2001 15.2, 15.4, 15.8, 15.18, 16.2, 17.2, 17.4, 17.10, 17.11.

Tuesday April 2, 2001 18.2.
Reminder: We have a test on Tuesday !

Tuesday April 10, 2001 No mandatory homework is assigned for next week. There is a bonus question: what do a and b have to satisfy if you want Z_n to be isomorphic to the direct product of Z_a and Z_b. For even more bonus points, prove your claim.

Tuesday April 17, 2001 24.8, 24.12, 24.13, 25.2, 26.4.
Bonus questions: Answer 25.2 for Z_n in general, 25.11, find a (left) zerodivisor in the ring of 2×2 real matrices.

Tuesday April 24, 2001 27.14, 27.23, 28.6, 28.7/d. Bonus question: Show that a²-2b² is not zero if a and b are integers and at least one of them is not zero.

Tuesday May 1, 2001 30.7.
Bonus question: 30.6: show that multiplication is compatible with the introduced equivalence relation. 31.13.

Date due	Problems:
Tuesday January 23, 2001	A.1/a,d,g,h,i, 1.2, 1.4, 1.24.
Tuesday January 30, 2001	3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 4.2 Bonus: 3.20, Prove that the relation " A is equivalent to B if there is a 1-1 and onto map from A to B" is an equivalence relation on sets.
Tuesday February 6, 2001	4.8, 5.6, 5.7, 5.8, 6.1/a,b,c,d, 6.2/a,c, 6.4, 7.2. Bonus: Prove that a map from S to S has a left inverse if and only if it is 1-1, and it has a right inverse if and only if it is onto.
Tuesday February 15, 2001	7.4, 7.2 (again!), 8.2, describe the composition of two central reflections. Bonus: 7.24.
Tuesday February 27, 2001	9.2, 9.9, 10.1, 10.12. Bonus: 10.16. Solve the Gear Puzzle I cited from the computergame Myst in class all show that it has no solution.
Tuesday March 13, 2001	11.2, 11.4, 11.6, 11.8, 12.1, 12.2. Bonus: Let r₁, r₂, r₃, and r₄ be the first four remainders in the Euclidean algorithm. Express them as a linear combination of a and b.
Tuesday March 20, 2001	13.2, 13.4, 13.6 (b), 13.11 (c) (draw also a picture indicating which divisor divides which other, as seen in class), 14.2, 14.4, 14.10.
Tuesday March 27, 2001	15.2, 15.4, 15.8, 15.18, 16.2, 17.2, 17.4, 17.10, 17.11.
Tuesday April 2, 2001	18.2. Reminder: We have a test on Tuesday !
Tuesday April 10, 2001	No mandatory homework is assigned for next week. There is a bonus question: what do a and b have to satisfy if you want Z_n to be isomorphic to the direct product of Z_a and Z_b. For even more bonus points, prove your claim.
Tuesday April 17, 2001	24.8, 24.12, 24.13, 25.2, 26.4. Bonus questions: Answer 25.2 for Z_n in general, 25.11, find a (left) zerodivisor in the ring of 2×2 real matrices.
Tuesday April 24, 2001	27.14, 27.23, 28.6, 28.7/d. Bonus question: Show that a²-2b² is not zero if a and b are integers and at least one of them is not zero.
Tuesday May 1, 2001	30.7. Bonus question: 30.6: show that multiplication is compatible with the introduced equivalence relation. 31.13.