Survey of Algebra. Math 3354, Section 3. Spring 2016

TuTh 2-3:15 pm, Cabell 323.

• Instructor: Mikhail Ershov
• Office: Kerchoff 302
• e-mail: ershov at virginia dot edu
• Office hours: 3 hrs TBA and by APPOINTMENT
• Course webpage:   http://people.virginia.edu/~mve2x/3354_Spring2016

References:

• required text: A book of Abstract Algebra, second edition, Charles Pinter
• additional text (recommended): Elements of Modern Algebra, eighth edition, Linda Gilbert
• online notes: preliminary version is 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 Sections 21-23 of Pinter's book, Chapter 2 of Gilbert's book and online lectures 1-9. The core part of the course (about 7 weeks) is concerned with group theory. The references for this part are Sections 3-5, 7-11, 13-16 of Pinter's book, Chapters 3 and 4 of Gilbert's book and online lectures 10-23. In the last 3-4 classes we will discuss selected topics from ring theory. The references for this part are Sections 17-20, 24-25 of Pinter's book, Chapters 5 and 6 of Gilbert's book and online lectures 24-27. I will assume familiarity with basic properties of sets and functions (the references for this part are Sections 2 and 6 of Pinter's book and Chapter 1 of Gilbert'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. MATH 3000 is offered this semester and will meet TuTh 9:30-10:45am. An assessment exam will be administered at the beginning of the semester to help you make the decision about MATH 3000 .

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 APMA 3080) 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 (this is covered, for instance, in Section 1.6 of Gilbert's book). 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 Jan 21 Introduction to algebraic structures. Jan 26, 28 Mathematical induction. Division Algorithm. Feb 2, 4 Divisibility. Greatest Common Divisor. Unique factorization theorem. Feb 9, 11 Congruences. Chinese Remainder Theorem. Feb 16, 18 Congruences classes. Feb 23, 25 Definition of a group. Examples and basic properties of groups. Mar 3 Subgroups. Mar 15, 17 Orders of elements. Cyclic groups. Isomorphisms. Mar 22, 24 Isomorphisms (continued). Homomorphisms. Mar 29, 31 Permutation groups I. Lagrange Theorem and classification of groups of small order. Apr 5,7 Cosets and Normal subgroups. Apr 12 Permutations groups II. Apr 19, 21 Quotient Groups. Apr 26, 28 Rings and ideals. May 3 Quotient rings.

Evaluation

The course grade will be based on homework, two midterms and the final (all in-class), with weights distributed as follows:
• 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: Tue, Mar 1st
• Second midterm exam: Thu, Apr 14th
• Final exam: Fri, May 6th, 9am-12pm

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 in-class exam or fail to submit a take-home exam by the due date 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 Thursday.
• No late homework will be accepted. However, three 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.