Math 371: Abstract Algebra

Syllabus, Winter 2007

Instructor: Lynn E. Garner Office: 348 TMCB Phone: 801-422-6153

Web page: http://www.math.byu.edu/~lynng/ e-mail: lynng@math.byu.edu

Text: Herstein, Abstract Algebra, 3^rd Edition, Chapters 1–4.

The Course: This is a first course in modern abstract algebra, which is the study of basic algebraic structures and their properties. In addition to abstract algebra being an important subject in its own right, it is also an essential tool in most other branches of mathematics. Thus abstract algebra is a core topic for all mathematics majors, mathematics education majors, and majors in many other disciplines that use higher mathematics and logic extensively.

Prerequisites: This course has both Math 190, Fundamentals of Mathematics, and Math 343, Linear Algebra, as prerequisites. Competence in the material of those courses will be assumed. In particular, the language of mathematics, including mathematical logic up through elementary predicate calculus and an understanding of proof and counterexample, will be necessary. Many examples will be drawn from linear algebra, such as matrix arithmetic. Most valuable to the student is the love of thinking about mathematical structures and concepts and reasoning through their inter-relationships.

Topics: The core topics of Math 371 include:

Basic set theory

Integers, the division and Euclidean algorithms, unique factorization

Mathematical induction

Real and complex numbers

Groups, basic definitions and examples

Subgroups, Lagrange’s theorem

Homomorphisms, homomorphism theorems

Normal subgroups and factor groups

Cauchy’s theorem

Direct products

Symmetric groups, even and odd permutations, cycle decompositions

Rings, basic definitions and examples

Ideals, ring homomorphisms, quotient rings

Maximal ideals

Polynomial rings, unique factorization in polynomial rings

Field of fractions of an integral domain

Objectives: The student should achieve mastery of the above topics, which means that the student is able to:

Give correct definitions

State correctly the major theorems, including hypotheses and limitations

Give examples and non-examples of various groups and rings

Solve problems relating to the above topics

Understand, create, and critique proofs of elementary results in these topics

Homework: Exercise sets will be assigned regularly and the solutions are typically due at the next meeting of the class. Each exercise set will be given a “batting average” score, and your total homework score will contribute 15% of your grade.

Homework Format: Please adhere to the following format. Use one side only of standard letter-sized paper. Put your name and the sheet number at the top of each sheet. Label each problem with its number and page and keep problems in order. Visually separate consecutive problems by drawing a line horizontally across the entire sheet between them. (If an exercise has lettered parts, each part is to be treated as a separate exercise.) Most solutions to problems will involve not only algebra but also explanations; use complete sentences and correct grammar. To submit homework, stack the sheets in order but do not fasten them together in any way; fold them lengthwise to form a “book” with the back of the last sheet on the outside. On the front of the “book” write your name, Math 371, and the Exercise Set number.

Midterm Tests: There will be three midterm tests, two on groups (including preliminary general topics) and one on rings. These will be in-class, closed-book exams, announced well in advance and consisting of problems similar to those in the exercise sets. Emphasis will be on using definitions and theorems rather than reciting them. Each midterm test will contribute 20% of your grade.

Final Exam: There will be a comprehensive final exam in the regular classroom on Monday, April 23, 2007, 7–10 am. It will be a closed-book exam and will be similar to the midterm tests. The final will contribute 25% of your grade.

Spirit of the Y: Since matters of faith rarely come up in mathematics (outside of Gödel’s theorem), your instructor wishes you to know that he fully supports the standards of conduct of the University and its sponsoring Church. He believes that in matters of secular learning, more important than what you learn is what you go through to learn it. He believes we should follow the injunction to seek learning, by study and also by faith, and that the diligence and obedience we show in this life in gaining knowledge and intelligence will give us an advantage in the life to come.

The Fine Print. Here are some statements included at the suggestion of the University that inform you of some of your legal rights and responsibilities relative to this class.

Preventing Sexual Harassment BYU's policy against sexual harassment extends not only to employees of the university but to students as well. If you encounter sexual harassment, gender-based discrimination, or other inappropriate behavior, please talk to your professor, contact the Equal Employment Office at 422-5895, or contact the Honor Code Office at 422-2847.

Students with Disabilities BYU is committed to providing reasonable accommodation to qualified persons with disabilities. If you have any disability that may adversely affect your success in this course, please contact the Services for Students with Disabilities Office at 422-2767. Services deemed appropriate will be coordinated with the student and instructor by that office.