## Math 113 - Sections 2 and 3 - Introduction to Abstract Algebra - Spring 2012

Prof. L. Williams - Tue. & Thu., 9:30-11am in Room 4 Evans (Section 2), and 3:30-5pm in Room 4 Evans (Section 3)

[Course goals] [Textbook] [Study tips] [Homework] [Exams and grading] [Syllabus] [Homework assignments]

ANNOUNCEMENTS:

#### Instructor: Prof. L. Williams (williams@math.berkeley.edu)

Office: 913 Evans.
My office hours: I will have office hours during RRR week during the following times:
Tuesday May 1, 4pm-5:30pm
Thursday May 3, 2:30pm-3:30pm.
GSI office hours: Semyon Dyatlov will be holding office hours during RRR week during the following times:
Monday April 30, 12pm-3pm in 740 Evans
Tuesday May 1, 1pm-4pm in 736 Evans
Wednesday May 2, 1pm-4pm in 939 Evans

### Course goals

Abstract algebra studies the structure of sets with operations on them. It aims to find general underlying principles common to the usual operations (addition, multiplication, etc.) on diverse sets such as integers, polynomials, matrices, permutations, and much more. We'll learn in particular about groups, rings, and fields.

A group is a set equipped with a binary operation satisfying certain axioms. Examples of groups include the integers with the operation of addition, or invertible n×n matrices with the operation of matrix multiplication, but there are many other examples (for instance, symmetries of geometric objects, permutations of finite sets, etc.). Thus, we will develop the general theory which describes common features of all groups.

A ring is a set equipped with two binary operations which satisfy certain properties similar to those of addition and multiplication. Examples of rings include integers, polynomials, or n×n matrices. A field is a ring with additional nice properties.

In addition to these specific topics, an important goal of the course is to acquire more familiarity with abstract mathematical reasoning and proofs in general, as a transition to more advanced mathematical courses.

### Textbook

The textbook for this course is: John Fraleigh, A First Course in Abstract Algebra, 7th edition, Addison-Wesley. This is a very readable book on the subject, containing lots of good exercises and examples.

Most of the lectures will correspond to particular sections of the book, and studying the book will be very helpful. However, material will often be presented in a different order or from a different perspective, and we'll occasionally discuss topics which are not in the book at all. Thus it is important to attend class and, since you shouldn't expect to understand everything right away, you are strongly encouraged to take notes.

### Study tips

As with any upper division math course, it is essential to thoroughly learn and understand the key definitions, remembering all the axioms. If you don't know exactly what a group is, then you have no hope of proving that something is or isn't a group.

In the same way it is necessary to learn the statements of the theorems; it is not necessary to memorize their proofs, however the more you understand and the better your command of the material will be. A useful study aid is to try and summarize the key ideas in the proof in a sentence or two;

The material in this course is cumulative (builds upon previous chapters) and gets somewhat harder, so it is essential that you do not fall behind.

A key to understanding is to ask your own questions. What is a good example? Why is such and such assumption necessary in a theorem, what happens if I drop it? Does this property imply that property, or is there a counterexample?

If you get stuck on something or are confused by a particular concept, you are encouraged to come to my office hours. I will be happy to discuss it with you. However, the more thought you have put into it beforehand, the more productive the discussion is likely to be.

Here is a handout created by George Bergman which contains some useful information on notations and conventions related to sets, functions, propositional logic, etc.

Here is a handout created by Elena Fuchs which contains some useful information on computing quotient groups.

### Homework

Homework is due every Thursday (with a couple of exceptions) at 9:40am sharp for those students in Section 2, and at 3:40pm sharp for those students in Section 3. You can either bring it to class or slide it under my office door. (If it doesn't fit under the door, please be more concise!). Assignments will be posted here a week before they are due.

No late homeworks will be accepted for any reasons, but it is OK if you miss the deadline once or twice because your lowest two homework scores will be dropped.

1. You are encouraged to discuss the homework problems with your classmates. The best way to learn is to think hard about a problem on your own until you get really stuck or solve it, then ask someone else how they thought about it. However, when it comes to writing down your solutions, you must do this by yourself, in your own words, without looking at someone else's paper or any other source.
2. Your answers should be written in complete sentences which explain the logic of what you are doing. For example, "x^2=4, x=2, x=-2" is not understandable: instead, write "since x^2=4, it follows that x=2 or x=-2". If your proof is unreadable it will not receive credit. Also, results of calculations and answers to true/false questions should always be justified. Proofs should be complete and detailed. The proofs in the book provide good models, but when in doubt, explain more. You can of course cite theorems that we have already proved in class or from the book.

There will be one in-class midterm, and a final exam. Students registered for my 9:30am class will take a final exam on May 9, 11:30am-12:30pm. Students registered for my 3:30pm class will take a final exam on May 11, 7:00pm-10:00pm. You must take the final exam for the class for which you are registered.

There will be no makeup exams. However the grading policy allows you to replace your midterm score with your final exam score. So in particular, if you miss the midterm, your final score will replace it.

There is no regrading unless there is an obvious error such as adding up the points incorrectly. Every effort is made to grade all exams according to consistent standards, so regrading one student's exam would be unfair to everyone else.

The course grade will be determined as follows: homework (dropping the two lowest scores) 20%; midterm 30%; final 50%. If you score higher on the final than on the midterm, your final score will replace your midterm score. Grades will be curved to a uniform scale before averaging.

### Syllabus

The following is the plan for the course. The schedule is only an approximation.

• Lecture 1, 1/17: About the course; sets, maps, cardinality, equivalence relations [section 0].
• Lecture 2, 1/19: Binary operations, isomorphism between binary operations [sections 2 and 3].
• Lecture 3, 1/24: Groups; examples: Z_n, matrix groups, group tables [section 4].
• Lecture 4, 1/26: Subgroups; examples including cyclic subgroups [section 5].
• Lecture 5, 1/31: Cyclic groups [section 6].
• Lecture 6, 2/2: Cyclic groups continued [section 6].
• Lecture 7, 2/7: Permutations, Cayley's theorem [section 8].
• Lecture 8, 2/9: More about permutations: orbits, cycles, alternating group [section 9].
• Lecture 9, 2/14: Cosets, Lagrange's theorem [section 10].
• Lecture 10, 2/16: Direct products. Fundamental theorem of finitely generated abelian groups (statement only) [section 11].
• Lecture 11, 2/21: Homomorphisms [section 13].
• Lecture 12, 2/23: Normal subgroups, quotient groups; the "fundamental homomorphism theorem" [section 14].
• Lecture 13, 2/28: Review for the midterm.
• 3/1: Midterm #1, in class. Covers material up to and including lecture on 2/23.
• Lecture 14, 3/6: More about normal subgroups and quotient groups.
• Lecture 15, 3/8: Commutator subgroup, abelianization, center [section 15].
• Lecture 16, 3/13: Group actions on a set, Burnside's formula [sections 16, 17].
• Lecture 17, 3/15: Rings and fields: basic definitions, examples [section 18].
• Lecture 18, 3/20: Integral domains [section 19].
• Lecture 19, 3/22: Fermat and Euler's theorems [section 20]; start fields of quotients of an integral domain [section 21].
• Lecture 20, 4/3: Field of quotients of an integral domain [section 21]; start rings of polynomials [section 22].
• Lecture 21, 4/5: Rings of polynomials; evaluation homomorphisms; division algorithm [section 22, beginning of section 23].
• Lecture 22, 4/10: Factorization of polynomials over a field; irreducible polynomials; Eisenstein's criterion; unique factorization in F[x] [section 23].
• Lecture 23, 4/12: Ideals and quotient rings [section 26].
• Lecture 24, 4/17: Prime and maximal ideals [section 27].
• Lecture 25, 4/19: Field extensions, minimal polynomials [section 29]
• Lecture 26, 4/24: Vector spaces over a field [section 30], "fundamental theorem of algebra" [section 31]
• Lecture 27, 4/26: Review for the final
• 5/9 11:30am-2:30pm, Moffitt Room 101: Final exam for students in the 9:30am class (covers the entire course).
• 5/11 7-10pm, McCone Room 141: Final exam for students in the 3:30pm class (covers the entire course).

### Homework Assignments

All assignments are due at the beginning of the class on the date stated below. No late homeworks are accepted.

• No homework due March 1st (first midterm).

### Homework Solutions

• The solutions to the problem sets will be posted on BSpace.