Mathematics 4321  Schedule/Homework
Spring 2007

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Homework for Test # 3

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Tuesday April 3

1. Study Section 42. Simple extensions

2. What is the difference between F[x] and F[a]?

3. Do problems # 1-5 on  Handout on Simple Extensions given in class

4. Question: Find a famous saying by Kronecker about the work of man.

Thursday April 5

1. Do problems # 3, 4, 5, 7, 11 on p. 197 Section 42.

2. Do problems # 6-21, 23, 24 on Handout on Simple Extensions

3. For extra challenge do problem # 25 on the Handout.

4. Read sections 43 and 44.

Homework # 4 due on Tuesday April 17: Problem # 42.7 and # 30 from Handout on Simple Extensions

Tuesday April 10

1. Study Section 43 Degrees of Extensions  and 44 Splitting Fields

2. Do Problems # 2, 4, 5, 7, 8 from Section 43 and Problems # 1-4, 11, 12, 13 from Section 44

3. For extra challenge do problems 43.19 and 44.16.

Thursday April 12

1. Study Section 45 Finite Fields

2. Do problems # 2, 4, 5.

Tuesday April 17

1. Study Section 46. Galois Groups

2. Compute Gal(E/F) where E=Q(\sqrt(3)), F=Q.

Thursday April 19

1.  Galois groups.

Tuesday April 24

Review Test # 3

Thursday April 26 Test # 3 (Covers Sections 42-46)

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Homework for Test # 2

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Tuesday February 20

1. Study  Chapter VIII. Section 34.Polynomials: Definition and elementary properties

2. Finish Handout # 3 given in class.

3. Do problems # 1-5, 8, 9.

4. For extra challenge do problems #6, 10, 11, 12

5. Question: The use of  "x" and other letters near the end of the alphabet to represent an "indeterminate" is
due to__________. He is also responsible for the first publication of the Factor Theorem in his work The Geometry,
which appeared as an appendix to his Discourse on Method.

Thursday February 22

1. Study Section 35. The division algorithm

2. Do problems # 1-10, 12, 13, 17.

3. What was the contribution of Girolamo Cardano to the solution of polynomial equations?

Tuesday February 27

1. Study Section 36. Factorization of Polynomials

2. Do problems # 1-12 all, 16.

3. For extra challenge do problems # 24.

Thursday March 1st

1. Study Handout on Factorization of Polynomials over a Field

2. Do the following problems: Section 44 # 5-10, Section 36 # 23, Handout # 10-21 and 23-26

3. For extra challenge do problems # 24 on p.173

Homework # 2 due on Thursday March 8: Problems # 34.10 and 25.25

Tuesday March 6

1. Study Section 38. Homomorphisms of rings. Ideals.

2. Do problems # 1, 5, 6, 7, 9, 10, 12, 16, 17, 22

3. For extra challenge do problems # 13, 18, 19

Homework # 3 due on Tuesday March 20: problems # 38.15

Thursday March 8

1. Study Section 39. Quotient rings

2. Do problems # 3, 4, 6, 7

3. For extra challenge do problems # 9, 10, 11

Monday-Friday March 12-16 SPRING BREAK!!!

Tuesday March 20

1. Study the definition of prime ideals and maximal ideals.

2. Study the proof of the theorems given in the handout.

3. Review all homework problems for in-class review on Thursday March 22.

Take-home portion of Test # 2 DUE THURSDAY APRIL 5th at 11:00 a.m.

1. Study Section 40. Quotient rings of F[x].
In particular study the proof of Theorem 40.1 and 40.3

2. Do problems # 1, 2

Thursday March 22

Review Test # 2

Tuesday March 27 Test # 2 (Covers Sections 34-36, 38, 39)

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Homework for Test # 1

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Tuesday January 16-First day of classes!

1. Study Section 24 Introduction to rings-understand all definitions and examples.

2. Read Handout-Amazing secrets to be successful in any mathematics advanced class! Write a summary!

3. Do problems # 1-6, 12, 15-18 pages 124-125.

4. For extra challenge do Problems # 19, 22.

5. Question: Who was the first mathematician to use the term "ring"?

Thursday January 18

1. Study Section 25-Integral domains. Subrings.

2. Do problems # 1-5, 9, 10, 12, 14, 17, 22 pages 127-128.

3. Give an example of a ring R with unity e that has a subring S with unity e' not equal to e.

4. For extra challenge do Problems # 16, 19, 23, 24 pages 127-128.

5. Question: Which mathematician is responsible for the development of axiomatic ring theory?

Tuesday January 23

1. Study Section 26-Fields

2. Do problems # 1-10, 14, 18, 19, 20, 23, 24 pages 130-131.

3. Finish the problems in the handout given in class.

4. For extra challenge do Problems # 12, 13, 16, 17 pages 130-131.

5. Question: Is it possible for the unity element in a subfield of a field to be different than the unity of the whole field?
Provide a proof or give a counterexample. Compare to Exercise 3 (Jan 18) above.

6. Read Section 27. Study the definitions of ring isomorphism and characteristic of a ring.

Thursday January 25

1. Study Section 27-Isomorphism. Characteristic

2. Do problems # 1-4, 9, 10, 13, 14, 15, 18, 19, 21 pages 134-135.

3. For extra challenge do Problems # 16, 23, 24 pages 134-135.

4. Homework # 1 due on Tuesday 1/30: Problem # 16 page 125; Problem # 22 page 128.

5. Read Sections 28 and 29.

Tuesday January 30

1. Study Section 28. Ordered integral domains and Section 29. The Integers

2. Do problems 1-6, 8, 9, 12 page 139.

3. Do problems 1, 3, 4, 6  page 141.

4. Read Section 30. Study the construction of Q.

Thursday February 1

1. Study Section 30. Field of Quotients.  The field of rational numbers.

2. Do problems # 8, 9 on page 145.

4. Question: True or false: If a is irrational, then 1/a is irrational.
Give examples to show that if a and b are irrational, then ab may be either rational or irrational, depending on a and b.

5. Study the definitions for Quiz on Tuesday

Tuesday February 6
Today Quiz on definitions!!

1. Study Section 31. Ordered fields. The field of real numbers.

2. Do problems # 3, 4, 7, 8, 9, 10, 11, 13, 17, 23, 25.

3. For extra challenge do problems #16, 18, 19 on page 149.

4. Read Section 32. Study the construction of C.

Thursday February 8

1. Study Section 32. The field of complex numbers

2. Do problems # 1-6, 11, 12, 13, 15

3. For extra challenge do problems #14, 17.

Tuesday February 13

Review for Test # 1

Thursday February 15-Test # 1  (Covers  Sections 24-32)