Survey of Algebra. Math 3354, Section 2. Spring 2015

TuTh 11am-12:15 pm, Rouss 410.

Text: Elements of Modern Algebra, eighth edition, 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 many of these from high school, but we will study them at a more sophisticated level. The core part of the course (about 8 weeks) is concerned with group theory (Chapters 3 and 4 and some material not covered in the book). In the last 3 classes we will discuss selected topics from ring theory (Chapters 5 and 6). 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 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 MWF 10-10:50am. 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 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
Jan 13, 15 2.1, 2.2 Introduction to algebraic structures. Mathematical induction.
Jan 20, 22 2.3, 2.4 Divisibility. Greatest common divisor.
Jan 27, 29 2.4, 2.5 Unique factorization theorem. Congruences.
Feb 3, 5 2.5, 2.6 Chinese Remainder Theorem. Congruence classes.
Feb 10, 12 3.1, 3.2 Definition of a group. Examples and basic properties of groups.
Feb 17, 19
First midterm, 3.3 Subgroups.
Feb 24, 26 3.3, 3.4 Subgroups (continued). Cyclic groups
Mar 3, 5 3.4, 3.5 Orders of elements. Isomorphisms.
Mar 17, 19 3.6, 4.1 Homomorphisms. Permutation groups I.
Mar 24, 26 4.4 Cosets and Lagrange Theorem. Classification of groups of small order.
Mar 31, Apr 2 4.5 Normal subgroups. Permutations groups II.
Apr 7, 9 Second midterm, 4.7     Direct sums and classification of finite abelian groups.
Apr 14, 16 4.6
 Quotient Groups.
Apr 21, 23 5.1, 6.1
 Rings and ideals.
Apr 30 6.2 Quotient rings.

Evaluation

The course grade will be based on homework, two midterms and the final (all in-class), with weights distributed as follows:

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.

Make-up policy

Homework

Collaboration policy on homework.

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: