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Homework assignments
(MATH 5/4161-001, Spring 2004)

Last update: April 27, 2004

Disclaimer: The information below comes with no warranty. If, due to typographical 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 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.

Notation: 1.2/1b means Exercise 1b in Section 1.2. The deadlines do not apply to the Bonus questions, which expire only once we solve them in class.

No. Date due: Problems:

13 We 04/28 No more homework will be assigned this semester. 10.2/1,2; 12.2/3 abd.
Bonus: 9.4/7b (you may use 7a).
Prove Theorem 12.3 using the fact that Gaussian integers have unique factorization.

12 Mo 04/19 9.3/3b,4; 9.4/2b.
Bonus: (3 pts) Find in the book a theorem stating a generalization of Theorems 9.3 and 9.8.

11 Mo 04/12 9.1/2; 9.2/1ace,2c,4b (using Euler's criterion, if necessary, but without using the Quadratic Reciprocity Law); 9.3/1ac, 5a (you have to use the Quadratic Reciprocity Law).

10 Mo 04/05 8.4/2b

9 Mo 03/29 8.1/2a; 8.2/1a,4a,8a; 8.3/1a,2a.

8 Mo 03/22 7.2/1,3,6,8; 7.4/2,5ac,8; 8.1/1c.
Bonus: Let H be a subgroup of a group G. Define a relation on the elements of G by setting a~b whenever ab^-1 belongs to H. Prove that you defined an equivalence relation.
Bonus: Prove or disprove that Euler's phi-function φ satisfies φ(n²)=nφ(n) for all positive integer n.

7 Fr 03/19 6.2/1a, 2, 4ac, 7b, 8a (use 6.2/3); 6.3/2c, 5a.

6 Mo 03/01 5.3/1,2a,6a; 5.4/1b,7; 4.4/4d, 7, 15c; 6.1/9.

5 Mo 02/23 4.3/1bc, 4a, 5; 4.4/1bc, 4a, 5.
Find the Lagrange interpolation polynomial p(x) satisfying p(-1)=5, p(0)=3, and p(1)=5.
Bonus: Given a prime p and a positive integer n, find the highest exponent k of p for which p^k divides n!. For half credit give a formula involving a sum using the integer part function, for full credit use the sum of digits of n in base p in your formula. [Mostly done on March 5, 2004; you may still try 6.3/7 for the quarter of the original bonus score.]

4 Mo 02/09 3.1/6bc, 16ab; 3.2/4ab, 5; 4.2/5, 8b, 9.
Bonus: We define the norm of a Gaussian integer a=a₁+a₂i as N(a)=a₁²+a₂². Prove that given any pair a,b of nonzero Gaussian integers, there are Gaussian integers q and r such that a=qb+r and N(r)<N(b).

3 Mo 02/02 Express the greatest common denominators in 2.3/1 as an integer combination of the inputs. (That is, write gcd(a,b) as ax+by for the given pairs (a,b).) 2.4/1ac, 2b.
Bonus: Prove that every prime is irreducible in an integral domain.

2 Mo 01/26 2.2/1,3,4a,12; 2.3/1, 4b.
Bonus: Consider the Euclidean Algorithm presented in the proof of Theorem 2.7. Express r₅ as a linear combination of a and b.

1 We 01/21 1.1/1b, 1c, 9, 1.2/1b, 5a, 3d, 1.3/1c, 2.1/3a.
Bonus: Prove the Second Principle of Induction for any well-ordered set (without reference to arithmetic operations).

No.	Date due:	Problems:
13	We 04/28	No more homework will be assigned this semester. 10.2/1,2; 12.2/3 abd. Bonus: 9.4/7b (you may use 7a). Prove Theorem 12.3 using the fact that Gaussian integers have unique factorization.
12	Mo 04/19	9.3/3b,4; 9.4/2b. Bonus: (3 pts) Find in the book a theorem stating a generalization of Theorems 9.3 and 9.8.
11	Mo 04/12	9.1/2; 9.2/1ace,2c,4b (using Euler's criterion, if necessary, but without using the Quadratic Reciprocity Law); 9.3/1ac, 5a (you have to use the Quadratic Reciprocity Law).
10	Mo 04/05	8.4/2b
9	Mo 03/29	8.1/2a; 8.2/1a,4a,8a; 8.3/1a,2a.
8	Mo 03/22	7.2/1,3,6,8; 7.4/2,5ac,8; 8.1/1c. Bonus: Let H be a subgroup of a group G. Define a relation on the elements of G by setting a~b whenever ab^-1 belongs to H. Prove that you defined an equivalence relation. Bonus: Prove or disprove that Euler's phi-function φ satisfies φ(n²)=nφ(n) for all positive integer n.
7	Fr 03/19	6.2/1a, 2, 4ac, 7b, 8a (use 6.2/3); 6.3/2c, 5a.
6	Mo 03/01	5.3/1,2a,6a; 5.4/1b,7; 4.4/4d, 7, 15c; 6.1/9.
5	Mo 02/23	4.3/1bc, 4a, 5; 4.4/1bc, 4a, 5. Find the Lagrange interpolation polynomial p(x) satisfying p(-1)=5, p(0)=3, and p(1)=5. Bonus: Given a prime p and a positive integer n, find the highest exponent k of p for which p^k divides n!. For half credit give a formula involving a sum using the integer part function, for full credit use the sum of digits of n in base p in your formula. [Mostly done on March 5, 2004; you may still try 6.3/7 for the quarter of the original bonus score.]
4	Mo 02/09	3.1/6bc, 16ab; 3.2/4ab, 5; 4.2/5, 8b, 9. Bonus: We define the norm of a Gaussian integer a=a₁+a₂i as N(a)=a₁²+a₂². Prove that given any pair a,b of nonzero Gaussian integers, there are Gaussian integers q and r such that a=qb+r and N(r)<N(b).
3	Mo 02/02	Express the greatest common denominators in 2.3/1 as an integer combination of the inputs. (That is, write gcd(a,b) as ax+by for the given pairs (a,b).) 2.4/1ac, 2b. Bonus: Prove that every prime is irreducible in an integral domain.
2	Mo 01/26	2.2/1,3,4a,12; 2.3/1, 4b. Bonus: Consider the Euclidean Algorithm presented in the proof of Theorem 2.7. Express r₅ as a linear combination of a and b.
1	We 01/21	1.1/1b, 1c, 9, 1.2/1b, 5a, 3d, 1.3/1c, 2.1/3a. Bonus: Prove the Second Principle of Induction for any well-ordered set (without reference to arithmetic operations).