Course Syllabus

Approximate Syllabus:

The following is a rough, tentative plan for the topics we will cover.

• Preliminaries. We will begin with a review of some essential preliminaries, including sets, functions, relations, induction, and some very basic number theory. You have probably already seen this material in Math 55 or elsewhere, so the review will be brief. Some of this material is in section 0 of the book, some is scattered throughout random later sections, some is in the notes on proofs, and some is in none of the above. 
• Groups. We will learn a lot about groups, starting with the detailed study of a slew of examples, and then proceeding to some important general principles. We will cover most of Parts I, II, and III of the book. We will mostly skip the advanced group theory in Part VII, aside from stating a couple of the results. (You can learn some of this material in Math 114.) We will completely skip Part VIII on group theory in topology; this material is best learned in a topology course such as Math 142. 
• Ring theory and polynomials. Next we will learn about rings. We will pay particular attention to rings of polynomials, which are very important e.g. in algebraic geometry. We will cover most of Parts IV,  and some of parts V, and IX. 
• Elements of field theory. Finally, after reviewing some notions from linear algebra in a more general setting, we will learn the basics of fields, from Part VI of the book. We will develop enough machinery to prove that one cannot trisect a sixty degree angle with a ruler and compass. We will not have time for the more advanced field theory in Part X, including the insolvability of the quintic; this is covered in Math 114.

Office Hours: Monday 9-12am, 1071 Evans Hall

Course Overview: https://math.berkeley.edu/~libland/teaching/math-113/