# Fall 2023

### Instructor

• Daniel Arreola, Office Hours: Tue and Thu 3–4:30pm in Carver 438.

### Time and Place

• Lectures: MWF 9:55pm–10:45pm in Carver 282

### Texts and Resources

• Main text: Hungerford, Abstract Algebra: An Introduction, 3rd Edition (available from the University Bookstore in both hardcopy and electronic format, and on Amazon)
• Supplementary text: Judson, Abstract Algebra: Theory and Applications (free online textbook)

The class will require weekly homework submissions, 2 in-class exams, and a final exam. Class components will be weighted as follows:

• Homework: 45%
• (Oct 2, 2023) Exam 1: 15%
• (Nov 6, 2023) Exam 2: 15%
• (Dec 13 at 7:30–9:30am) Final Exam: 25%

Your homework should be neat and stapled. Problems should be listed in order and proofs given in complete sentences with correct notation. Use of LaTeX is encouraged but not at all required. For each homework question you must:

• Write out the statement of the question – ideally you will paraphrase the given question, highlighting key points while removing any irrelevant details.
• Explain your solution with a complete and logical succession of ideas. Each problem will be graded on the following 4 point scale:

4=Very Good - Correct mathematics that is carefully thought out and thoroughly explained with complete sentences.

3=Good - Correct mathematics with some minor gaps in explanation.

2=Basic - Partial solution with errors in understanding or computation.

1=Emerging - Work that has some merit but also has significant shortcomings in the mathematics or explanation.

0=No Credit - No work submitted or no serious attempt.

### Homework

Homework sets will be posted here. They are due most Wednesdays. First homework is due August 30.

To get you going, here is what GPT-4 had to say about your first homework problem: Non-commutative function composition (Is it correct?)

1. (Due Aug 30) Appendix B: 9, 11bd, 12bc, 24abdf, 25bd, 26cd, 28; 1.1: 2(a)(b), 9; Homework 1
2. (Due Sep 6) 1.2: 1dh, 4, 22, 24; 1.3: 14, 22, 30, 34, 36; Homework 2, LaTeX file
3. (Due Sep 13) 2.1: 6, 12, 14, 16, 20, 22; Appendix C: 15, 17; Appendix D: 11, 17; Homework 3, LaTeX file
4. (Due Sep 20) 2.2: 6, 8, 12, 14(c), 16; 2.3: 2, 6, 10; 3.1: 2, 4; Homework 4, LaTeX file
5. (Due Sep 27) 3.1: 12, 22, 30, 42; 3.2: 12, 18, 22, 33, 40; Homework 5, LaTeX file
6. (Due Oct 11) 3.3: 12(a)(b)(d)(e), 19, 30, 42; 4.1: 5(b)(d), 6(a)(c)(d)(e), 14, 16; 4.2: 5(b)(d), 14; Homework 6, LaTeX file
7. (Due Oct 18) 4.3: 6, 12, 14, 22; 4.4: 2(a)(d), 8(c)(d)(e)(f), 10, 16; Homework 7, LaTeX file
8. (Due Oct 25) 4.5: 4, 8, 18(a)(b), 19(b); 4.6: 1(b), 4; 5.1: 4, 8, 10, 12; Homework 8, LaTeX file
9. (Due Nov 1) 5.2: 2, 6, 14, 16; 5.3: 1, 2, 6, 8; Homework 9, LaTeX file
10. (Due Nov 15) 7.1: 4, 8, 10, 24, 30; 7.2: 7(b)(d), 12, 20, 24, 34; Homework 10, LaTeX file
11. (Due Nov 29) 7.3: 6, 12, 14, 28; 7.4: 6, 12, 14, 24; Homework 11, LaTeX file
12. (Due Dec 6) 7.5: 3, 4, 6, 14, 21, 22; 8.1: 6, 20, 26, 29, 33; Homework 12, LaTeX file

### Lecture Summary

We will cover Appendix A,B,C,D and Chapters 1–5, 7 and Section 8.1. A short summary of each lecture will be posted here.

1. Appendix A - Logic and Proof
2. Appendix B - Sets and Functions
3. 1.1 - The Division Algorithm
4. 1.2 - Divisibility: The greatest common divisor
5. 1.2 - Divisibility: Euclid's Algorithm
6. 1.3 - Primes and Unique Factorization
7. (Labor day)
8. 2.1 - Congruence in $$\mathbb{Z}$$
9. Appendix C - Well-ordering and Induction
10. Appendix D - Equivalence Relations and 2.2 - Modular Arithmetic
11. 2.3 - Structure of $$\mathbb{Z}_n$$
12. 3.1 - Definition and Examples of Rings
13. 3.1 - (continued)
14. 3.2 - Basic Properties of Rings
15. 3.2 - (continued)
16. 3.3 - Isomorphisms and Homomorphisms
17. 3.3 - (continued)
18. Review for Exam 1
19. (Oct 2) Exam 1
20. 4.1 - Polynomial Rings $$R[x]$$
21. 4.2 - Divisibility in $$F[x]$$
22. 4.3 - Irreducibles and Unique Factorization
23. 4.4 - Polynomial Functions, Roots, Reducibility
24. 4.4 (contd.) and 4.5 - Irreducibility in $$\mathbb{Q}[x]$$
25. 4.5 (contd.)
26. 4.6 - Irreducibility in $$\mathbb{R}[x]$$ and $$\mathbb{C}[x]$$
27. 5.1 - Congruence in $$F[x]$$
28. 5.2 - Arithmetic in $$F[x]/(p(x))$$
29. 5.3 - Structure of $$F[x]/(p(x))$$
30. extra/review/examples of $$F[x]/(p(x))$$
31. 7.1 - Definition and Examples of Groups
32. 7.2 - Basic Properties of Groups
33. Review for Exam 2
34. (Nov 6) Exam 2 on Sections 3.3, 4.1–4.6 and 5.1–5.3.
35. 7.2 - (contd.)
36. 7.3 - Subgroups
37. 7.3 (contd.) Center and Cyclic Groups
38. 7.4 - Isomorphisms and Homomorphisms
39. 7.4 - (contd.)
40. (after break) 7.5 - The Symmetric and Alternating Groups
41. 7.5 – 8.1 - Congruence and Lagrange's Theorem
42. 8.1 - (contd.)
43. Extra/Review
44. Extra/Review
45. Extra/Review

### Statement on Free Expression

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.