MATH 3010 Abstract Algebra

Fall 2024

Instructor

• jthartwig.net
• Office 470
• Email jth@iastate.edu
• Office hours: MWF 3:10pm–4:00pm

Time and Place

• Lectures: MWF 11:00pm–11:50pm in Carver 128

Texts and Resources

• Main text: Algebra — Notes from the Underground by Paolo Aluffi

Exams and Grading

• Homework: 40%

• Exam 1: 20% (September 27, 2024) Chapters 1–3 Study Guide for Exam 1

• Exam 2: 20% (November 8, 2024) Chapters 4, 5, and 7 Study Guide for Exam 2

• Final Exam: 20% (Thursday December 19, 2024, 7:30am–9:30am in Carver 128)

• In-Class Worksheets: 10%

• Worksheet 1

Homework

Homework sets will be posted here. They are due most Wednesdays.

1. (Due Sep 4) Homework 1
2. (Due Sep 11) 1.4, 1.5, 1.8, 1.9, 1.10, 1.11, 1.12, 1.15, 2.1, 2.6
3. (Due Sep 18) 1.25, 1.26, 2.2, 2.5, 2.8, 2.9, 2.12, 2.13, 2.15, 2.16
4. (Due Sep 25) 2.19, 2.22, 3.1, 3.2, 3.3, 3.5, 3.10, 3.11, 3.12, 3.17
5. (Due Oct 9) 3.15, 3.19, 3.22, 4.1, 4.2, 4.6, 4.7, 4.8, 4.10, 4.17
6. (Due FRIDAY Oct 18) 4.18, 4.19, 5.2, 5.3, 5.4, 5.5, 5.6, 5.8, 5.9 (three parts, each graded as a separate problem), 5.15
7. (Due Oct 23) 5.19, 5.20, 5.24, 5.25, 5.27, 5.32, 5.36 (each problem worth 7 pts)
8. (Due Oct 30) 6.10, 7.2, 7.3, 7.5, 7.8, 7.9, 7.14, 7.18, 7.19, 7.33 (\(p\le 20\) is sufficient)
9. (Need not be handed in) 4.4, 4.9, 4.17, 5.2, 5.6, 5.28, 7.21, 7.25, 7.29, 7.30 Solutions not covered in class
10. (Due Nov 20) 11.3, 11.4, 11.8, 11.10, 11.11, 11.12, 11.15, 11.16, 11.17, 11.18
11. (Due Dec 4) 11.20, 11.21, 11.22, 11.24, 11.26, 11.27, (1) Find the order of each element of the dihedral group of order \(10\). (2) For each element in the dihedral group of order \(10\), write its permutation representation as a product of disjoint cycles.
12. (Due Dec 11) 11.17 (an automorphism of \(G\) is an isomorphism from \(G\) to itself), 11.37, 11.42, 12.1, 12.9, 12.12

Lecture Summary

We will cover Appendix A.5–A.8, and Chapters 1–5, 7, 11, and parts of Chapter 12. A short summary of each lecture will be posted here.

1. Set operations (A.5), functions (A.6, A.7)
2. Compositions, identity (A.6, A.7); relations, equivalence classes (A.8)
3. §1.1 Well-Ordering Principle and Induction
4. (Labor Day)
5. §1.2 Division with Remainder
6. §1.3 Greatest Common Divisors
7. §1.4 The Fundamental Theorem of Arithmetic
8. §2.1 Equivalence Relations and Quotients, §2.2 Congruence mod \(n\)
9. §2.3 Algebra in \(\mathbb{Z}/n\mathbb{Z}\), §2.4 Properties of + and \(\cdot\) on \(\mathbb{Z}/n\mathbb{Z}\)
10. §2.4 (continued) Zero-divisors and multiplicative inverses in \(\mathbb{Z}/n\mathbb{Z}\). Fermat’s Little Theorem
11. §2.4 (continued) RSA encryption algorithm, §3.1 Definition and examples of rings
12. §3.2 Basic Properties
13. §3.3 Special Types of Rings
14. Review
15. Exam #1 Friday September 27, during lecture
16. §4.1 Cartesian Products of Rings
17. §4.2 Subrings
18. §4.3 Ring Homomorphisms
19. §4.4 Isomorphisms of Rings
20. §5.2 Kernel and Ideals
21. Worksheet 1
22. §5.3 Quotient Rings
23. §5.4-§5.5 The First Isomorphism Theorem
24. §7.2 Polynomial rings over fields
25. §7.2 (continued)
26. §7.2 (continued) \(F[x]\) is a PID
27. §7.3 Irreducibility in Polynomial Rings
28. §7.4 Irreducibility in \(\mathbb{Q}[x]\) and \(\mathbb{Z}[x]\)
29. §7.4 (continued)
30. §7.5 Irreducibility Tests in \(\mathbb{Z}[x]\)
31. §7.5 (continued), and Review
32. Review
33. Exam #2 Friday November 8, during lecture
34. §11.1 Groups: definition, examples, first properties
35. §11.1 (continued)
36. §11.1-§11.2 Symmetric Group, Group Actions
37. §11.2-§11.3 Group Actions and Homomorphisms to \(S_n\); Symmetries of regular polygons
38. §11.3 Description of the Dihedral Group
39. §11.3 Order, cyclic groups; even/odd permutations, cycles and transpositions
40. §11.4 Cosets
41. §12.1 Lagrange’s Theorem, §11.4 Normal Subgroups
42. (planned) §11.4 Quotient Groups (Maybe Worksheet #2)
43. (planned) §11.5 The First Isomorphism Theorem for Groups
44. (planned) Review
45. (planned) Review

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