MT 3310: Introduction To Abstract Algebra

Overview: This course is concerned with abstract algebra. This is one of the most fundamental areas of modern mathematics. The idea is to study mathematical structures with simple rules---so simple that they may be found in many different areas---and yet to see that these simple rules impose a rich structure upon the objects. Much of the course will be concerned with Group Theory. Groups are behind games such as the Rubrik's cube, and they are also behind the study of symmetries in physics and secure data transmission in computer science. They play a central role in modern mathematics. We will also study objects called Rings. These objects mimic the integers, and yet may be subtly different. Their properties will allow us to discover new facts about the integers themselves. Finally, we will study Fields. These are Rings with extra structure. They are important in understanding the solutions to equations and are also used in coding theory and in cryptography.

This is a foundational upper division course covering one of the subjects which is at the center of modern mathematics. But it is intrinsically sophisticated stuff. You can't get by just studying right before the test. You need to go over your notes after each lecture. Learn the definitions by heart. Work through the examples (in both text and lecture) until you understand them thoroughly. Then learn the proofs as well as the Theorems. If you put this off, you'll find that the lectures make use of material you have not yet worked through, and you'll get behind rapidly. But if you keep up---and don't hesitate to ask questions in class or in office hours if you're confused by something---then the course will work well. The material is intrinsically beautiful, and you'll be introduced to the power of modern abstract mathematics.

Note: This course satisfies the algebra requirement for the B.A. degree in mathematics. Students considering the B.S. degree should take MT3311 and not this course.

Lectures: MWF 1:00-1:50 in GASSON 210.

Instructor: Dubi Kelmer, Maloney Hall 526. Email address: dubi.kelmer@bc.edu

Office Hours: MW 11-12 and Th 1-2.

I am very happy to answer your questions and to help you with the homework during these office hours. These times are always reserved for you and there is no need for an appointment. I am also available on other times by appointment. Please email me for one. I am always happy to work with students, so don't hesitate to contact me (email is also good for short questions).

Course Syllabus in PDF

Textbook: Abstract Algebra: A First Course by Dan Saracino, Second Edition. Waveland Press

Grading: Homework 30%; Each midterm 20%; Final 30%.

Exams: Two in-class exams (Wed. Feb. 10 and Wed. Mar. 23) and one final exam (Wed. May 11 at 12:30).

Homework: There will be weekly homework assigned and collected every week.

HOMEWORK WILL NOT BE ACCEPTED AFTER THE DAY IT IS DUE!

You are encouraged to typeset your HW in LATEX. Though it takes a while to get the hang of it, this is recommended as it will help you write mathematics clearly. If you want to give it a try, here are some helpful resources

Mac OSX:

Download MacTeX. For an introduction, see "Trying out TeX" and its links.

Windows: Download proTeXt and follow the installation instructions.

See also LaTeX manual (math symbols on page 60) and LaTeX Wiki Book for more information

Homework problems:

• Assignment 1: Problems 0.4, 0.9, 0.16, 0.17, 0.18, 1.3, 1.4, 1.5, 1.6, 1.9 from the text. Due Friday Jan 29 in class.
• (If you want to typeset your solution, you can use the following TEX template).
• Assignment 2: Problems 2.1, 2.4b,2.5,2.8, 2.10, 3.4, 3.5, 3.9, 3.10, 3.11, 3.12, and 3.15 from the text. Due Friday Feb. 5 in class. (Bonus problem: 3.16).
• Assignment 3: Problems 4.1, 4.5, 4.10, 4.13, 4.14, 4.20, 4.22, 4.25 from the text. Due Monday Feb. 15 in class.
• Assignment 4: Problems 5.1 (a,d,e,f,g), 5.5, 5.6, 5.7, 5.9, 5.14, 5.16, 5.18, 5.25 from the text. Due Friday Feb. 19 in class.
• Assignment 5: Problems 6.1c, 6.2, 6.3 (prove your answer), 6.7, 6.12, 6.14, 7.7,7.8 from the text. Due Friday Feb. 26 in class.
• Assignment 6: Problems 8.2ab, 8.3, 8.4, 8.7, 8.16, 8.17, 8.18 from the text. Due Friday Mar. 03 in class.
• Assignment 7: Problems 8.20, 8.23, 9.1 (if not, explain why), 9.4,9.5,9.9, 9.12, 9.16 from the text. Due Friday Mar. 18 in class.
• Assignment 8: Problems 10.1, 10.2, 10.5, 10.6, 10.7, 10.8, 10.16, 10.19, 10.21 from the text. Due Friday Apr. 1st in class.
• Assignment 9: Problems 11.1, 11.2, 11.3, 11.4, 11.7, 11.16, 11.19, 11.23, 11.24 (Bonus problem 11.32) from the text. Due Friday Apr. 8 in class.
• Assignment 10:Problems 12.1abcd, 12.3, 12.4abcdef, 12.8, 12.13, 12.15, 12.21, 13.1, 13.5, 13.7, 13.14, from the text. Due Friday Apr. 15 in class.
• Assignment 11:Problems 16.1, 16.3, 16.9, 16.13, 16.14, 16.15, 16.16, from the text. Due Monday Apr. 25 in class.
• Assignment 12:Problems 17.1 (explain your answer), 17.6, 17.9, 17.14, 17.27, 17.33, 18.1, 18.2, 18.3 from the text. Due Monday May. 2 in class.