Math 430: Applied Algebra
Homework & Lecture Schedule
Homework assignments will be posted at least one week before the due date. It is your responsibility to check the course page on Piazza, which you can also easily access by logging into Blackboard, to obtain the assigned problems.You are expected to start working on the homework sets early (NOT the day they are due or right before). It is extremely difficult to answer last-minute homework questions; particularly if you have not been participating in the Piazza discussion beforehand.
Help with writing up assignments
To improve your mathematical writing quickly, start by writing draft solutions to homework early. A day or two later after you have had time to forget what you wrote, read it. If it doesn’t make sense or convince you, rewrite it. Writing a solution requires saying what you mean and meaning what you say. Be intellectually honest. Intellectual dishonesty includes: 1) stating a “reason” without understanding its relevance. 2) Claiming a conclusion when you know you haven’t proved it. 3) Giving an example and claiming you have proved the statement for all instances.(This text borrowed from Prof. Kaul)
Lecture schedule
You are expected to cover (at least at a high level) the assigned readings before coming to the lecture. This will help you follow the course and organize your notes.Homework problem sets are naturally related to the material covered in the course; hence, homework numbers are listed next to the corresponding topic.
| Dates | Tentative topics covered | Assigned reading | Related homework |
| August 24&26 | Topic 1: What is applied algebra? - a few minutes of introduction to the course. Preliminaries: Equivalence relations; posets; functions; cardinality. |
Book: Chapter 1. | Homework 1, due 9/3. |
| September 1&3 | Week 2: (finishing up preliminaries with cardinality and) The integers: Well-ordering principle, congruences, division in the integers. GCD, Euclidean algorithm. |
Book: 1.4, 2.1, 2.2. | Homework 2, due 9/10. |
| September 8&10 | Week 3: (finishing up properties of integers:) Congruences. The Chinese remainder theorem. | Book: 2.3, |
Homework 3, due 9/17. |
| September 15&17 | Week 4: Introduction to groups: permutations of a set; binary operations: semingroups and monoids. | Book: 3.1, start of 3.2 | Homework 4, due 9/24. |
| September 22&24 | Week 5: Introduction to groups (continued): symmetry group; groups and subgroups. | Book: second half of 3.2, 3.3. | Homework 5, due 10/1. |
| September 29 & Oct1 | Week 6: Cyclic groups, order of an element; Cosets, Lagrange's Theorem, |
Book: finish 3.3; cover 4.1, and start 4.2 if possible. | Homework 6, due 10/8. |
| October 6 and 8: | Week 7: Lagrange's Theorem and consequences; subroups of cyclic groups; lattice of subgroups and what it encodes; group isomorphisms; order of a subgroup. | Book: section 4.1 and some topics from 3.3 revisited (e.g., lattice of subgroups). | Homework 7, due 10/22 (!) |
| Midterm exam: October 13th | Topics covered: homework sets 1-5, book chapters 1,2 and 3. | NO hw due this week, due to exam. | |
| October 15: | Week 8: topics: remaining results from 4.1. that are 'useful in computation with groups'; introduction to normal subgroups. | Book: tie up loose ends from 4.1, start 4.2. | Homework 7, due 10/22 (!) |
| October 20&22: | Week 9: topics: normal subgroups, center, commutator, quotient groups. Direct products. Homomorphisms. Image and kernel. | Book: 4.2 and start of 4.3. | Homework 8, due 10/29 |
| October 27&29: | Week 10: topics Image and kernel. The isomoprhism theorems. Automorpshisms. Exact sequences. * In-class QUIZ is planned this week. |
Book: finish 4.3 (skip last couple topics). | Homework 9, due 11/5 |
| November 3&5: | Week 11: Introduction to rings; subrings, ideals, homomorphisms, and isomorphism theorems. We will start chapter 6! | Book: 6.1 and 6.2. | Homework 10, due 11/12 |
| November 10&12: | Week 12: Integral domains, division rings, and fields. Maximal and prime ideals.
Division in rings: Euclidean domains. * If there is a take-home part of the final exam, it will be announced this week, and assigned shortly thereafter. |
Book: 6.3; 7.1 (select topics). | Homework 11, due 11/19 |
| November 17&19: | Week 13: Euclidean Domains, PIDs. | Book: 7.1, 7.2, 7.3 (select topics). | Homework 12, due 11/24*** (This is a Tuesday!) |
| November 24: | Week 14: topics PIDs, UFDs. | Book: 7.3. and 7.4. | Homework 14, due 12/1*** (This is a Tuesday!) |
| December 1&3: | Week 15: topics Roots of polynomials, splitting fields. Hint as to what field theory looks like. The first isomorphism theorem, kernel of the canonical homomorphism, and quotient rings and ideals in the context of splitting fields and irreducible polynomials. | Book: 7.4. | No remaining homework sets; focus on final exam. |
Final Exam covering Chapters 4, 6, and 7. |
Monday, December 7, 8-10 a.m., E1-106. If there is a take-home part of the final exam, it will be announced at least 4 weeks in advance, and assigned at least one week before it is due (12/7). |