MTH 331/431 Abstract Algebra I (Fall 2021):
- What is Abstract (or Modern) Algebra?
Algebra is defined to be the study of algebraic structures. Mathematicians study algebraic structures from a general point of view, compare different structures, and find relationships between them. In this course we will study elementary number theory, groups, rings, and fields.
Find more information about the course here:
MTH 331 syllabus
MTH 331&H syllabus (honors)
MTH 431 syllabus (graduate) - Instructor: Dr. Sofya Chepushtanova, office SLC 408, email: sofya.chepushtanova@wilkes.edu.
- Meeting Times: MWF 11:00-11:50 am and T 03:00-03:50 pm, SLC 359.
- Office Hours: TF 10:00-10:50 am, MWF 12:00-12:50 PM, or by appointment.
- Textbook: Tom Judson's Abstract Algebra: Theory and Applications.
An electronic version of the book is freely available here. - Sage: an open-source program for doing mathematics and is the ideal companion to Tom Judson's textbook. Sage includes many mature and powerful open-source tools for mathematics, such as GAP, a system for computational discrete algebra (used for group theory). With a strength in number theory, Sage also has excellent support for rings and fields. We will be using Sage sometimes for exercises and assignments. More information about Sage is here and here.
- Reading:
- LaTeX: a markup language to typeset documents. It excels at making math and the overall layout beautiful.
You can use it to type your homework solutions. Read the tutorial here. You may find the following LaTeX cheat-sheet useful.
Consider the following LaTeX template to start: download (it is the template of document On Proofs).
Schedule:
Week of |
Class Topics | Remarks, Materials, Assignments |
8/30 | Introduction Preliminaries (Ch. 1) |
Introduction slides Sets and Equivalence Relations |
9/6 | No class on 9/6 (Labor Day) Integers (Ch. 2) |
Induction and Well-Ordering Principle Division Algorith, Euclidean Algorithm, Prime Numbers Homework 1 due 9/10: All students: Ch. 1, §1.4 "Exercises" (online text), exer. 7, 9, 18(b,c), 19, 20(b), 21, 22(c,e), 25(d). Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 26, 28, and 29. Latex template for Homework 1: here. Homework 1 Solutions |
9/13 | Groups (Ch. 3) | Integer Equivalence Classes and Symmetries Homework 2 due 9/17: All students: Ch. 2, §2.4 "Exercises" (online text), exer. 2, 5, 12, 15(d, f), 18. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 16 and 23. Latex template for Homework 2: here. Homework 2 Solutions |
9/20 | Groups Continued Subgroups (Ch. 3) Cyclic Groups (Ch. 4) |
Groups: Definitions and Examples Group examples handout Subgroups Homework 3 due 9/24: All students: Ch.2, §2.4, exer. 31; Ch. 3, §3.5, exer. 1 (b,d), 2 (c,d), 6, 7, 8, 10, 14. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 28 in Section 2.4, and 15 and 26 in Section 3.5. Latex template for homework: here. Homework 3 Solutions |
9/27 | Exam I on 9/27 (Monday) Exam I solutions: MTH 331 version MTH 331&/431 version Cyclic Groups Continued |
Cyclic Groups Subroups of the Circle Group Study for Exam I: study guide Homework 4 due 10/1: All students: Ch. 3, §3.5, exer. 16, 31, 34, 39, 43, 48. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 46 and 47. NOTE: bonus problem for all - exer. 45. Latex template for homework: here. Homework 4 Solutions |
10/4 | Permutation Groups. (Ch. 5) |
Permutation Groups Explore Sage sections 1.6, 2.7, 3.8, and 4.8. Homework 5 due 10/8: All students: Ch. 4, §4.5, exer. 1 (a,b,c), 2 (a,b,d,f), 3 (b,c,h,k), 4 (d), 11, 23 (a), 32. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 1 (e), 23 (c), and 26. NOTE: bonus problem for all - exer. 31. Latex template for homework: here. Homework 5 Solutions |
10/11 | Permutation Groups Continued. Dihedral Groups. Cosets and Lagrange's Theorem. (Ch. 6) No class on Friday (Fall Recess) |
Dihedral Groups Homework 6 due 10/18: All students: Ch. 4, §4.5, exer. 6; Ch. 5, §5.4, exer. 1, 2 (b,d,g,h,j,p), 3 (b,d), 4, 22, 23, 24, 27. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 13 and 35 in Section 5.4. NOTE: bonus problems for all: 6 and 34 in Section 5.4. Latex template for homework: here. Homework 6 Solutions |
10/18 | Cosets and Lagrange's Theorem Cont'd. Fermat's and Euler's Theorems. Isomorphisms. (Ch.9) |
Cosets and Lagrange's Theorem Fermat's and Euler's Theorems Homework 7 due 10/29 (new deadline!): All students: §6.5, exer. 1, 3, 4, 5 (a,b,c), 11 (only prove statements (a) and (d) are equivalent), 17. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 18, 19. NOTE: bonus problems for all: 5 (f), 13. Latex template for homework: here. Homework 7 Solutions |
10/25 | Exam II on 10/25 (Monday) Exam II solutions: MTH 331 version MTH 331&/431 version Isomorphisms Cont'd. |
Study for Exam II: study guide .
Isomorphisms Topics for presentations Homework 8 due 11/8: All students: §9.4, exer. 2, 7, 8, 9 (do not need to proof that G is a group, prove isomorphism only), 16 (b and d), 34, 46, 50. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 24 and 41. NOTE: bonus problems for all: 15, 25. Latex template for homework: here. Homework 8 Solutions |
1/11 | Direct Products. (Ch.9) Normal Subgroups and Factor Groups. (Ch.10) |
Direct Products |
11/8 | Factor Groups Continued. Homomorphisms. (Ch.11) |
Normal Subgroups. Factor Groups. Simple Groups. Homomorphisms Homework 9 is due 11/15: All students: §10.4, exer. 1 (a,b,c,e), 5, 6, 8, 13 (a,c). Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 9, 11. NOTE: bonus problems for all: 13 (b,d). Latex template for homework: here. Homework 9 Solutions |
11/15 |
The Isomorphisms Theorems. (Ch.11) Finite Abelian Groups (briefly). (§13.1 - first part) Rings. (Ch.16) |
The Isomorphisms Theorems Finite Abelian Groups Homework 10 is due 11/30: All students: §11.4, exer. 2 (b,c,d,e), 4, 9, 10, 12, 13 and §13.4, exer. 2 and 3. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 16 and 18 in §11.4. NOTE: bonus problems for all: exer. 7 (a,b,c) in § 11.4. Latex template for homework: here. Homework 10 Solutions |
11/22 | Exam III on 11/22 (Monday) Exam III solutions: MTH 331 version MTH 331&/431 version Tuesday follows Thursday schedule, no class! Thanksgiving Break: 11/24-11/28 |
Study for Exam III: study guide |
11/29 |
Rings Continued. Integral Domains and Fields. (Ch.16)
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Rings Integral Domains and Fields Ring examples handout Homework 11 is due 12/10: All students: §16.7, exer. 1 (b,e), 3 (b,d), 8, 9, 18 (c,d), 23. Honors (MTH 331&H) and graduate (MTH 431) students: in addition, do exer. 2 and 31. NOTE: bonus problems for all: exer. 1(h) and 36. Latex template for homework: here. Homework 11 Solutions |
12/6 | Chapter on Rings Cont'd. Student presentation: Everard Riley on Structure of Groups |
Ring Homomorphisms and Ideals |
12/13 | Monday follows Friday schedule! Student presentation: Cordell Siggins on Cryptography My office hours during the finals: Tuesday, Wednesday, Thursday: 9:30 - 11:00 am |
Take-home Final Exam is due 12/20/21: click here |