In the last hundred years or so, "algebra'' has come to mean the study of abstract structures involving operations on sets, modeled on the operations that arise from a variety of concrete geometric and combinatorial situations. In almost every branch of mathematics this kind of algebra seems to emerge as the main technical toolset which encodes the mechanics, if not the essence, of the material at hand. Math 113 is intended as an introduction to this toolset, its main applications, and the methods of mathematical proof and problem solving. After a brief warmup with the familiar numbers, sets, and functions we will turn to the main objects of "modern algebra'': groups, rings, and fields. I will try to emphasize the connections among these ideas and other branches of mathematics, as well as geometric and dynamic ways of thinking about them. Students will be expected to ask questions in office hours, class, and by email.. As a text, I will use the third edition of * Abstract Algebra, *by John Beachy and William Blair. This is a very readable text, and students will be expected to learn a good bit of the material directly from it.. My role is to point out what I feel are the most important topics, to explain and elaborate key or unclear points in the text, and to answer questions. I can do this best if I hear from students ahead of time what may be causing them difficulty, and so I encourge them to contact me by email (ogus@math.berkeley.edu) with questions about the text, preferably before the topic is scheduled for a lecture. I will post a syllabus, which I will update and modify as the course progresses. Be sure to consult this regularly and to read the material in the text before the corresponding lecture.

The course will be graded in a serious manner, based on weekly homework assignments, at least one midterm, and the final exam. There may also be unannounced quizzes at random times. We are in Exam group 5, so our exam will be Friday, May 16, at 12:30 pm, in 289 Cory.

- A thorough mastery of the material, including the main definitions, the major theormes and their proofs, as well as a demonstration of originality in solving problems and writing proofs.
- Good understanding of the material, including the main definitions, theorems, and proofs, and ability to solve nontrivial problems.
- Firm grasp of the main points, including the major definitions and theorems, ability to solve standard problems.
- Familiarility with major concepts, terminology, and problem solving technqiues.
- None of the above.

My grades mean the following:

I will be assisted by a Graduate Student Instructor, Arturo Prat-Waldron, whose office hours will be Wednesdays, 9 am--2 pm, and Thursday 9--11 am and 3:30--6:30 pm, in 891 Evans. He will help with the homework assignments and answer general questions; of course your are strongly encouraged to attend my own office hours as well.