MTH 331/431 Abstract Algebra I (Fall 2019):
- 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 in the syllabus. - Instructor: Dr. Sofya Chepushtanova, office SLC 410, email: sofya.chepushtanova@wilkes.edu.
- Class meetings: MWF 11:00-11:50am, room SLC 411 and R 1:00-1:50pm, SLC 378.
- Office Hours: MWF 9:00-9:50am, MW 2:00-2:50am, or by appointment, room SLC 410.
- 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 |
8/25 | Introduction. Preliminaries. (Ch. 1) |
Introduction notes Sets and Equivalence Relations |
9/1 | No class on 9/2 (Labor Day) Integers. (Ch. 2) |
Induction and Well-Ordering Principle Division Algorith, Euclidean Algorithm, Prime Numbers Homework 1 due 9/6: All students: Ch. 1, §1.3 "Exercises" (online text), exer. 7, 9, 18(b,c), 19, 20(b), 21, 22(c,e), 25(d). Graduate (MTH 431) students: in addition, do exer. 26 and 29. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 1 Solutions |
9/8 | Groups. (Ch. 3) | Integer Equivalence Classes and Symmetries Homework 2 due 9/13: All students: Ch. 2, §2.3 "Exercises" (online text), exer. 2, 5, 12, 15(d, f), 18. Graduate (MTH 431) students: in addition, do exer. 17(c) and 23. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 2 Solutions |
9/15 | Groups Cont'd. Subgroups. (Ch. 3) |
Groups: Definitions and Examples Summary of Some Groups Examples Subgroups Homework 3 due 9/20: All students: Ch.2, §2.3, exer. 28, 31; Ch. 3, §3.4, exer. 1 (b,d), 2 (c,d), 6, 7, 8, 14. Graduate (MTH 431) students: in addition, do exer. 10 and 15 in Section 3.4. (MTH 331 students are welcome to do these problems for extra credit.) Latex template for homework: here. Homework 3 Solutions |
9/22 | Exam I on 9/23 (Monday) Exam I solutions: MTH 331 version MTH 431 version Cyclic Groups. (Ch. 4) |
Cyclic Groups Study for Exam I (9/23): study guide Homework 4 due 9/27: All students: Ch. 3, §3.4, exer. 16, 26, 33, 39, 43, 48. Graduate (MTH 431) students: in addition, do exer. 34 and 47. (MTH 331 students are welcome to do these problems for extra credit.) NOTE: Extra problem for all: 45. Latex template for homework: here. Homework 4 Solutions Sage Lab on Thursday: explore "Sage" Sections 1.5, 2.6, and 3.7. |
9/29 | Cyclic Groups Cont'd. Permutation Groups. (Ch. 5) |
Subroups of the Circle Group Homework 5 due 10/7 (Monday): All students: Ch. 4, §4.4, exer. 1 (a,b,c), 2 (a,b,d), 3 (b,c,h,k), 4 (d), 11, 23 (a), 31. Graduate (MTH 431) students: in addition, do exer. 2 (f), 23 (c), and 26. (MTH 331 students are welcome to do these problems for extra credit.) NOTE: Extra problem for all: 32. Latex template for homework: here. Homework 5 Solutions |
10/6 | Permutation Groups Cont'd. Fall Recess 10/10 and 10/11: no classes! |
Permutation Groups Dihedral Groups Homework 6 due 10/16: All students: Ch. 4, §4.4, exer. 6; Ch. 5, §5.3, exer.1, 2 (b,d,g,h,j,p), 3 (b,d), 4, 22, 23, 24, 27. Graduate (MTH 431) students: in addition, do exer. 13 in Section 5.3. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 6 and 34 in Section 5.3. Latex template for homework: here. Homework 6 Solutions |
10/13 | Exam II on 10/17 (Thursday) Exam II solutions: MTH 331 version MTH 431 version Cosets and Lagrange's Theorem. (Ch. 6) |
Study for Exam II (10/17): study guide
Homework 7 due 10/25: All students: §6.4, exer. 1, 3, 4, 5 (a,b,c), 8, 11 (only prove statements (a) and (d) are equivalent), 17. Graduate (MTH 431) students: in addition, do exer. 18, 19. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 5 (f), 13. Latex template for homework: here. Homework 7 Solutions |
10/20 | 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 |
10/27 | Isomorphisms Cont'd. Direct Products. (Ch.9) |
Isomorphisms Direct Products Homework 8 due 11/6: All students: §9.3, exer. 2, 3, 7, 8, 9 (do not need to proof that G is a group, prove isomorphism only), 16 (b and d), 34, 46, 50. Graduate (MTH 431) students: in addition, do exer. 24 and 41. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 15, 25. Latex template for homework: here. Homework 8 Solutions |
11/3 | Normal Subgroups and Factor Groups. (Ch.10) Homomorphisms. (Ch.11) |
Normal Subgroups. Factor Groups. Simple Groups. Homework 9 is due 11/13: All students: §10.3, exer. 1 (a,b,c,e), 5, 6, 8, 9, 13 (a,c). Graduate (MTH 431) students: in addition, do exer. 11 and 12. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: 13 (b,d). Latex template for homework: here. Homework 9 Solutions |
11/10 | Exam III solutions: MTH 331 version MTH 431 version The Isomorphisms Theorems. (Ch.11) |
Homomorphisms The Isomorphisms Theorems Study for Exam III (11/14): study guide |
11/17 | Finite Abelian Groups. (§13.1 - first part) Rings (Ch.16) |
Finite Abelian Groups (briefly) Rings Homework 10 is due 11/25: All students: §11.3, exer. 2 (b,c,d,e), 4, 9, 10, 12, 13, 18; §13.3, exer. 2 and 3. Graduate (MTH 431) students: in addition, do exer. 16 in §11.3. (MTH 331 students are welcome to do this problem for extra credit.) Extra problems for all: exer. 7 (a,b,c) in § 11.3. Latex template for homework: here. Homework 10 Solutions |
11/24 | Chapter on Rings Cont'd. (Tuesday follows Thursday schedule) Thanksgiving Recess: no classes 11/27-29 |
Integral Domains and Fields Homework 11 is due 12/9: All students: §16.6, exer. 1 (b,e), 3 (b,d), 4(b,d,e), 8, 9, 18 (c,d), 23. Graduate (MTH 431) students: in addition, do Exercises 2 and 16. (MTH 331 students are welcome to do these problems for extra credit.) Extra problems for all: exer. 1(h) and 5. Latex template for homework: here. Homework 11 Solutions |
12/1 | Chapter on Rings Cont'd. Student presentations: Paul Gladstone, Landon Henry - 12/6 |
Ring Homomorphisms and Ideals Maximal and Prime Ideals (OPTIONAL) Student presentations: list of topics |
12/8 | (Monday follows Friday schedule) Student presentations: Julia Rostron, Jessica Roxby - 12/9 Final Exams start 12/10/19 |
Take-home Final Exam is due 12/16/19: click here |