Math 113: Abstract Algebra

Instructor: Alex Kruckman

Email: kruckman@math.berkeley.edu

Office Hours: Monday-Thursday 9:30-10:30am in 775 Evans (7th floor).

Meetings: Monday-Thursday, 2:10-4:00pm in 75 Evans (ground floor).

The way the course appears on schedule.berkeley.edu is a bit confusing. We are scheduled to meet 2:10-4:00 four days a week (not just Tuesday and Thursday), and in practice there will not be a distinction between "lecture" and "discussion" (but see the Class Participation section below).

This is a condensed summer course, and we will move extremely quickly.
Attendance is strongly advised!

Course Website: /~kruckman/summer2013/

Textbook: Fraleigh, A First Course in Abstract Algebra, 7th Edition

Different editions of this book vary widely - make sure you have the 7th Edition.

Most lectures will correspond to sections of the textbook, but I will not cover everything in the same order or from the same perspective as Fraleigh. I recommend that you read the textbook carefully as well as keeping careful notes - often multiple expositions of the same subject can greatly aid understanding. The textbook is quite well-written, but you should not expect it to be an easy read. To get the most out of math books, you have to be prepared to really engage, go slowly, and think carefully about what you are reading.

Content: Math 113 is an introduction to groups, rings, and fields. These are fundamental algebraic structures, which you can think of as generalized number systems. They abstract the operations of addition and multiplication present in number systems such as the integers, the real numbers, the complex numbers, etc. and allow us to make precise analogies between the algebra of these systems and others: polynomials, rational functions, matrices, permutations, symmetries in geometry, etc.

The study of groups, rings, and fields originated in the problem of the (un)solvability of equations by radicals, which was famously resolved by Évariste Galois in the early 19th century. Although we will not have time to study Galois theory in detail, my goal will be to develop enough building blocks to sketch a proof of the unsolvability of quintic equations at the end of the course.

An equally important goal of Math 113 is to develop your skills at creating and communicating mathematical arguments. This is probably among the first courses you have taken in which the majority of time - in lecture and on the homework - will be spent proving things. We will talk about proof strategies and style in class, but the most important thing to remember is that proofs are written to be read: clarity and precise use of language are just as important as logical correctness.

Grading: Your grade will consist of 8 homework assignments, a midterm exam, a final exam, and a classroom participation component.

10% - Classroom Participation
30% - Homework (schedule and assignments on the main course webpage)
30% - Midterm Exam (in class, Thursday 7/18)
30% - Final Exam (in class, Thursday 8/15)

There are no predetermined cutoffs for letter grades. I will let you know how homework and exam scores correspond to letter grades as we go along.

Classroom Participation: On a regular basis (typically the first half of class on Mondays and Wednesdays), I will give you problems to be completed in groups. After each group has solved their assigned problem, one member of the group will be asked to explain the solution to the rest of the class at the blackboard.

To receive full credit for classroom participation, you are expected to participate in your group discussions and present at least twice by the end of the summer. Your grade will not be based on correctness of the solution you present.

Homework: Homework is perhaps the most important part of the course: the best way to become familiar with new mathematical ideas is to use them, and the best way to get better at thinking mathematically is to prove things.

Because the course moves so quickly, no late homework will be accepted. It's okay if you don't manage to solve every problem on every assignment! Your homework will be graded partially for correctness and style (on a few problems), and partially for completion. You won't receive detailed feedback on every problem, but I will post solutions to the homework assignments on the course website.

Collaboration on the homework is encouraged, with the following caveats:
1. After a collaborative discussion in which a problem is solved, each student should then write up their solution individually and from scratch. This part of the process is of key importance for learning to communicate mathematically - and often it is while writing down an argument carefully that the key components crystallize in your mind.
2. Make sure that all participants in a collaboration are engaged in the discussion. You will not learn nearly as much by watching other students solve your problems or by splitting up the work.

On a related note, please do not scour the internet for homework solutions or ask for help on internet forums. If you are stuck, discuss with other students and come to office hours.

Exams: Exams will be 2 hours (or rather 1 hour and 50 minutes) long, held in class on Thursday July 18th and Thursday August 15th. Please make sure now that you can attend the exams. No make-up exams will be given, except for serious medical reasons.

Exam questions will be similar in style and difficulty to homework questions. The final exam will focus on material from the second half of the course (although certainly material from the first half of the course will be relevant).