Math 113: Abstract Algebra

General Information

• Place: 289 Cory Hall
• Time: 10AM-12PM, Mon-Thu
• Dates: June 20th - August 12th
• Instructor: James McIvor, 1095 Evans Hall
• Email Address: my last name at math.berkeley.edu
• Office Hours: Mon 3-5; Tue 1-3; Wed 1-2
• Piazza Course Discussion

• Group Theory
(updated 7/15)
• Ring/Field Theory
(updated 8/9)

• Here is a practice exam for the final. Here is the solution.
• Note change in Monday Office Hours - now 3-5. Hoping to get a larger room for office hours, too - more updates later.
• Please Read "Preliminary Materials" under "Miscellaneous", below, as well as the appendices in Goodman's text, if you need a review of sets, injective/surjective functions, etc.

Overview

The course is divided into two halves, the first of which studies groups, the second of which studies rings and fields. All three of these are fundamental algebraic structures which are familiar to you already, but we will study them more systematically and theoretically than perhaps you have in the past.

Groups should be thought of as symmetries, and as such often have nice geometric interpretations which I shall emphasize heavily. For example, the set of all possible rotations of a circle forms a group, as does the set of symmetries of a cube. So we will draw many circles and cubes (and squares and tetrahedra and...).

Rings and fields tend to come in two flavors - "number systems" and sets of functions. The integers, the rational numbers, the real numbers, number systems in which 5=0 (?!) all are rings, as are the sets of continuous functions on the real line, differentiable functions, etc. They all have in common that you can add, subtract, and multiply. In cases where you are allowed to divide as well, the ring is called a field. So the rational numbers are a field, as are some number systems in which 5=0, but the continuous functions on the real line are not, since the quotient of two continuous functions may no longer be continuous. Rings and fields also have deep connections to geometry, though not necessarily as symmetries of geometric objects.

The course culminates in a brief exposure to Galois Theory, which was a legendary achievement largely responsible for the language and style of modern algebra. It shows a surprising connection between groups and fields. For example, you will learn how the polynomial x cubed minus two is related to a triangle and its symmetries.

Prerequisites

Textbook

Grading

You must take the second exam to pass the course - this is a strongly enforced university policy. In the event of a serious medical emergency, you may miss the first exam if you have written documentation of your illness. In that case the second exam will count for 200 points.

Homework

In doing your homework, you should work with others, but please write the names of your collaborators at the top of the assignment. You must write clearly - points will be taken off if your explanations are confusing or illegible. When writing proofs, you must use complete sentences. There will be eight assignments. Three problems will be selected for grading, out of five points, so each assignment is worth 15 points. I will drop your lowest score and replace it with your highest score.

Quizzes

Exams

Lectures