Syllabus for Math 3354.

Survey of Algebra. Math 3354, Section 1. Fall 2024

MW 2-3:15 pm, New Cabell 323.

Instructor: Mikhail Ershov
Office: Kerchof 302
e-mail: ershov at virginia dot edu
Office hours: 3 hrs TBA and by APPOINTMENT
Course webpage

References:

official text: Abstract Algebra: An Introduction , third edition, Thomas Hungerford
online notes: preliminary version available here

Course outline

This course introduces basic structures of modern algebra: groups, rings and fields. Approximately the first 4 weeks of the course will be devoted to elementary number-theoretic topics: greatest common divisor, unique factorization theorem, congruences etc. You should be familiar with many of these from high school, but we will study them at a more sophisticated level. The references for this part are Chapters 1 and 2 of Hungerford's book and online lectures 1-9. The core part of the course (about 6 weeks) is concerned with group theory. The references for this part are Chapters 7 and 8 of Hungerford's book and online lectures 10-23. In the last 2-3 weeks we will discuss selected topics from ring theory. The references for this part are (portions of) Chapters 3-6 of Hungerford's book and online lectures 24-26.

I will assume familiarity with basic logic and basic properties of sets and functions (see Appendices A and B of Hungerford's book). You should read this material on your own in the first 2-3 weeks and ask me questions about unclear points.

A note on proofs. MATH 3354 is a proof-based course. Everything we discuss in class will be rigorously proved. More importantly, you are expected not only to understand proofs, but also to learn how to construct your own proofs and how to write proofs (so that others can understand your argument). It is not expected that you have taken a proof-based course before MATH 3354. However, MATH 3354 will not include any lectures devoted specifically to proof writing; instead you will be expected to develop this skill gradually as we progress through the material. If you feel that you need a more detailed introduction to proof writing, you should consider taking MATH 3000 (Transition to Higher Mathematics) before or concurrently with MATH 3354. This semester MATH 3000 will meet MWF 10-10:50am + Th 5-5:50. An assessment exam will be administered at the beginning of the semester to help you decide whether taking MATH 3000 might be beneficial for you.

A note on linear algebra. The only official prerequisite for this course is MATH 1320. However, you are strongly encouraged to take linear algebra (MATH 3351 or equivalent) before or concurrently with MATH 3354. The only formal linear algebra skill that will be needed in MATH 3354 is the ability to add and multiply matrices. However, some of the mateiral in MATH 3354 is based on the ideas, which also appear in linear algebra, but in less abstract setting. Thus, having taken linear algebra may help you better understand some of the topics in MATH 3354.

Preliminary schedule.

week	topics
Aug 28	Introduction to algebraic structures.
Sep 2, 4	Mathematical induction. Division Algorithm.
Sep 9, 11	Divisibility. Greatest Common Divisor. Unique factorization theorem.
Sep 16, 18	Congruences. Congruence classes
Sep 23, 25	More on Congruences classes. Definition of a group.
Sep 30, Oct 2	Examples and basic properties of groups. Subgroups.
Oct 7, 9	First midterm. Orders of elements. Cyclic groups.
Oct 16	Isomorphisms.
Oct 21, 23	Homomorphisms. Direct products. Permutation groups I.
Oct 28, 30	Lagrange Theorem and classification of groups of small order. Cosets.
Nov 4, 6	Normal subgroups. Permutations groups II.
Nov 11, 13	Second midterm. Quotient groups.
Nov 18, 20	Quotient Groups (continued). Rings.
Nov 25	Ideals.
Dec 2, 4	Polynomial Rings. Quotient rings.

Evaluation

The course grade will be based on homework, two midterms and the final with weights distributed as follows. The current plan is that all exams will be given in-class, but this has not been finalized.

homework: 15%
midterms: 20% each
final: 45%

Exams

The midterm dates given below are tentative and may be changed later. The date and time of the final exam is determined by the registrar and cannot be changed.

First midterm exam: Mon, Oct 7th
Second midterm exam: Mon, Nov 10th
Final exam: Tue, Dec 10th, 2-5pm

Make-up policy

Make-ups will be given only under extreme circumstances (such as serious illness). Except for emergencies, you must obtain my permission for a make-up before the exam.
If you miss an exam without a compelling reason, you will be assigned the score of 0 on that exam.
University regulations specifically prohibit early make-ups.

Homework

Homework will be assigned weekly and will usually be due on Thursdays
Homework should be submitted electronically on Canvas. Please submit the entire homework as a single pdf and (if you original work is handwritten) make sure the file is readable; you are strongly encouraged to use scanning software (or a scanner) instead of making photos of individual pages.
No late homework will be accepted. However, two lowest homework scores will be dropped.
GRADING: it will not be possible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit.

Collaboration policy on homework.

You are welcome (and even encouraged) to work on homework together, but you must write up solutions independently, in your own words. In particular, you should not be consulting others during the process of writing down your solution .

Announcements

Major announcements will be made in class and also posted on the course webpage. Some other announcements may only be made by e-mail, so check your e-mail account regularly.

Add/drop/withdrawal dates:

Tuesday, Sep 10 -- Last day to ADD a course, elect the audit option or change the grading option (grade or CR/NC)
Wednesday, Sep 11 -- Last day to DROP a course (deletion from the transcript)
Tuesday, Oct 22 -- Last day to withdraw from a course

SDAC

All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.