Math 354, Section 2. Survey of Algebra.

Survey of Algebra. Math 3354, Section 2. Fall 2010

TuTh 11am-12:15 pm, Cabell 431.

Instructor: Mikhail Ershov
Office: Kerchoff 302
e-mail: ershov at virginia dot edu
Office hours: Tue 4:15-5:45, Wed 2:15-3 and 5-5:45 and by APPOINTMEN

Text: Elements of Modern Algebra, seventh edition, Jimmie Gilbert and Linda Gilbert.

Course outline

This course introduces basic structures of modern algebra: groups, rings and fields. Approximately first 4 weeks of the course will be devoted to elementary number-theoretic topics (Chapter 2 of the book): greatest common divisor, unique factorization theorem, congruences etc. You should be familiar with most of these from high school, but we will study them at a more sophisticated level. The core part of the course (about 7 weeks) is concerned with group theory (Chapters 3 and 4). In the last 3 weeks we will discuss selected topics from ring theory (Chapters 5 and 6) and (if time allows) some additional topics, possibly not covered by the book (in the latter case, additional references or online notes will be provided). I will assume basic familiarity with material of Chapter 1 (mostly set theory). You should read this chapter on your own in the first couple of 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 detalied introduction to proof writing, you may consider taking MATH 3000 (Transition to Higher Mathematics) before or concurrently with MATH 3354. MATH 3000 is not offered this semester, but is expected to be offered in Spring 2011.

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 our 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	sections	topics
Aug 24, 26	2.1, 2.2	Axioms for basic systems of numbers. Mathematical induction.
Aug 31, Sep 2	2.3, 2.4	Divisibility. Greatest common divisor.
Sep 7, 9	2.4, 2.5	Unique factorization theorem. Congruences.
Sep 14, 16	2.6, 3.1	Congruence classes. Definition of a group.
Sep 21, 23	3.1, 3.2	Examples and basic properties of groups.
Sep 28, 30	3.3, 3.4	Subgroups. Cyclic groups.
Oct 5, 7	First midterm, 3.4	Orders of elements.
Oct 14	3.5	Isomorphisms.
Oct 19, 21	4.1, 4.4	Permutation groups. Cosets.
Oct 26, 28	4.4, 4,5	Lagrange Theorem. Normal subgroups.
Nov 2, 4	4.6.	Quotient groups.
Nov 9, 11	Second midterm, 4.7	Direct sums.
Nov 16, 18	5.1, 6.1	Rings and ideals.
Nov 23	6.2	Quotient rings.
Nov 30, Dec 2	5.2	Integral domains and fields.
Dec 7	6.3	The characteristic of a ring.

Exams

First midterm exam: Tue, Oct 5th, in class;
Second midterm exam: Tue, Nov 9th, in class.
Final exam: Thu, Dec 16th, 9am-12 pm, Cabell 431.

Evaluation

Final exam: 45%
Midterm exams: 20% each
Homework: 15%

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 policy

Homework will be assigned weekly. The assignments will be posted on the course webpage and will usually be due on Thursday in class (due dates may be occasionally changed, e.g. during midterm weeks).
Please STAPLE your homework.
No late homework will be accepted. However, three lowest homework scores will be dropped.
You are 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.
GRADING: due to the large class size, it will not be posssible to grade all homework problems. In each assignment 3-4 selected (but not announced in advance) problems will be graded for credit. You may also ask for additional problems in your assignment to be graded (for feedback only).

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, September 7 -- Last day to ADD a course, elect the audit option, change the grading option (grade or CR/NC), or establish an independent study
Wednesday, September 8 -- Last day to DROP a course (deletion from the transcript)
Tuesday, October 19 -- Last day to withdraw from a course (instructor's permission required)