## Abstract Algebra (Math 434), Spring 2003

### Homework Exercises

Chapter Page Computational Theoretical Rotman
1
12 234 2, 12, 20, 40, 45 18, 22, 27, 29, 43, 48 5, 7, 8
13 246 6, 11, 19 (see 16), 20 (see 13), 35, 60 14 (see 13), 24, 41, 46, 54, 55 (see 23) 10, 11, 13, 16
14 260 5, 6ab, 22, 27, 29, 30, 34, 35, 42 (see 40), 45 10, 32, 39, 53 31, 33, 37
15 277 5, 6, 12, 15, 20, 21, 38 29, 30, 37, 40, 44, 54, 56 27, 29
16 290 12, 13, 18, 22, 23 4, 8, 9, 21, 24, 30, 31, 40 40, 45
17 307 8, 10, 11, 14, 21, 23 12, 19, 25, 30, 33 53, 58, 64, 65
18 325 13, 14, 17, 19, 21 8, 10, 12, 27, 31, 32, 35
19 339 5, 6, 13, 15, 22 2, 8, 11, 19, 24, 27, 29, 30, 31 68, 69, 71
20 357 1-5, 7, 8, 9,13, 25, 26 21 None
21 369 8, 12, 14, 16, 24, 26 2, 4, 7, 9, 11, 18, 23 None
22 381 1, 6, 11, 20, 21, 26 5, 10, 18, 25
23 389 3, 4, 5, 7 8, 9, 14 Read Appendix C
32 560 2, 10, 11, 12, 18 7, 8, 23, 25

1. Make your subject line exactly, exactly as follows: Math 434 Chap X, where X is the chapter covered. If you do not do this exactly right, your mail will not get filtered, and will not get graded. Also, please do not send me messages on other topics that have subject lines that begin with "Math 434" or they WILL get filtered and will not be read until I have occasion to grade some reading questions.
2. Put your full name as the first line of the body of your message.
3. Answer the questions in order, beginning each with the problem number.
4. Answers are due at 10:00 in the evening on the day before we begin discussing each new Chapter (usually Sunday night). They will not be accepted late.

[Chapter 12][Chapter 13][Chapter 14] [Chapter 15][Chapter 16][Chapter 17] [Chapter 18]

[Chapter 19][Chapter 20][Chapter 21] [Chapter 22][Chapter 23][Chapter 32]

Chapter 12

1. How are rings and groups alike?
2. How are rings and groups different?
3. Give an example of a noncommutative ring.
4. Give an example of a ring that would be of especial interest to a student contemplating a career in secondary mathematics education.
5. The direct sum construction is reminiscent of which construction for groups? Why?

Chapter 13

1. Give an example of a zero divisor in a ring of matrices.
2. In the same ring you used in the previous answer, give an example of non-identity unit.
3. What is the characteristic of a ring?
4. Give an example of an integral domain with characteristic 6.
5. Give an example of a ring, different from your answer in the previous question that has characteristic 6.

Chapter 14

1. How are ideals like normal subgroups?
2. How are ideals different from normal subgroups?
3. State the definition of a prime ideal.
4. State the definition of a maximal ideal.
5. Suppose R is a ring and A is an ideal. When is R/A an integral domain? Similarly, when is R/A a field?

Chapter 15

1. How are ring homomorphisms different from group homomorphisms?
2. How are ring homomorphisms like group homomorphisms?
3. What is the most important thing about the kernel of a ring homomorphism?
4. What is the precise meaning when we say that a field contains a "copy" of Zp or Q?
5. What part of this section is of the greatest interest to prospective elementary school teachers?

Chapter 16

1. What is a monic polynomial?
2. What is a PID?
3. What is a primitive root of unity?
4. What is the remainder for the division (x5+1)/(x2+3x+1)?
5. Which result from this section is of the greatest interest to prospective high school teachers?

Chapter 17

1. Define an irreducible polynomial.
2. What is the nth cyclotomic polynomial and why might we care?
3. Factor x4 + 3x3 + 2x + 4 over Z5.
4. Factor x3 + 3x + 2 over Z5.
5. Factor x4 - 2x2 + 8x + 1 over Q (the rationals).

Chapter 18

1. State the definition of an irreducible element in an integral domain.
2. State the definition of a prime element in an integral domain.
3. In what types of rings are prime elements and irreducible elements the same?
4. List the classes of principal ideal domains, unique factorization domains, and Euclidean domains as subsets of each other.
5. Can you give an example of a PID that is not an ED?

Chapter 19

1. How is a vector space like a group?
2. How does a vector space differ from a field?
3. What is a subspace?
4. State the definition of linear independence.
5. Why does Theorem 19.1 precede the definition of dimension?

Chapter 20

1. Define an extension field.
2. Define a splitting field.
3. How many splitting fields can there be for a polynomial p(x)?
4. Give an example of an irreducible polynomial over the rationals that has a root of multiplicity 2.
5. What is a perfect field?

Chapter 21

1. What is the difference between an algebraic element and a transcendental element?
2. What is the degree of an extension field?
3. Define the minimal polynomial of an algebraic element.
4. State a fundamental theorem that relates degrees of extensions in a tower of three fields.
5. Summarize the importance of the Primitive Element Theorem.

Chapter 22

1. Why are Galois fields named after Galois?
2. How would you build a finite field of order 83,521?
3. How would you build a finite field of order 143?
4. How many subfields of order 49 are there in a field of order 2401?
5. How many subfields of order 343 are there in a field of order 2401?

Chapter 23

(Today's questions brought to you by Hellekson, Willard and Lerman)
1. List the three tools you get while performing geometric constructions.
2. Can you construct a 120-sided regular polygon with these tools?
3. Can you construct a 144 sided regular polygon with these tools?
4. Give an example of an angle that can be trisected.
5. Why were these construction impossibilities not solved until the nineteenth century?

Chapter 32

1. Why is the Fundamental Theorem of Galois Theory called a "labor-saving device"?
2. What important problem can be solved using Galois Theory?
3. What does it mean to be solvable by radicals?
4. State the Feit-Thompson Theorem. Why is it "monumental"?
5. Give an example of a non-solvable group.

Rob Beezer, BEEZER(at)UPS(dot)EDU