Math 113: Abstract Algebra

University of California, Berkeley. Spring 2016.

MWF 3:10-4:00 PM in Room 9, Evans Hall.### Instructor:

Martin Helmer#### GSI Office Hours:

There is also a GSI for all the Math 113 sections. The GSI will hold drop-in office hours on a regular basis for you to go and get help on problems or answers to general questions about the material. The GSI for Math 113 is George Melvin.

- Tuesday: 4-6 PM
- Wednesday: 9-11 AM, 5-7 PM
- Thursday: 5-7 PM
- Friday: 9-11 AM

### Book:

Abstract Algebra: Theory and Applications by Judson (2015 Edition)

### Course Goals:

The goal of this course is to introduce the study of abstract algebra and for students to gain an understanding and appreciation of the elegance, utility and mathematical importance of several algebraic structures; specifically groups, rings and fields. It is hoped through the course of the class that students will come to see how these algebraic structures allow us to see common structure and behaviour between diverse sets such as the integers, polynomials, matrices, etc.

A group is a set with a binary operation satisfying certain axioms. Examples of groups include the integers with the operation of addition and square invertible matrices with the operation of matrix multiplication (and many others). A ring is, roughly speaking, a group with an additional operation, this can be thought of as having an operation analogous to addition and an operation analogous to multiplication. Examples include the integers, and polynomials. A field can be thought of as a ring with additional properties, roughly speaking a field has inverses to elements under the operation analogous to multiplication. Examples of fields include the real numbers, the complex numbers and the rationals.

By studying the abstract properties of these objects we will gain an understanding of a wide variety of mathematical objects and will illustrate the importance and utility of thinking about mathematics in terms of both abstract structure and concrete examples. This course will also give students the opportunity to acquire more familiarity with abstract mathematical reasoning and proofs in general, which will be important for future mathematical courses.

### Course Schedule and Notes:

We will cover three general algebraic structures in this course, these are: groups, rings and fields.

**Introduction**

- Week 1 (Jan. 20, 22):

**Groups**

- Week 2 (Jan. 25, 27, 29):
- Week 3 (Feb. 1, 3, 5):
- Week 4 (Feb. 8, 10, 12):
- § 5.1, 5.2: The Alternating Group, Dihedral Groups
- § 6.1, 6.2: Cosets, Lagrange's Theorem
- § 6.2, 6.3: Lagrange's Theorem, Fermat's and Euler's Theorems

- Week 5 (Feb. 17, 19):
- § 9.1: Isomorphisms (Definition and Examples)
- § 9.1, 9.2: Isomorphisms, Direct Products

- Week 6 (Feb. 22, 24, 26):
- § 9.2: Direct Products
- § 10.1, 11.1 [Justin Chen]: Factor Groups and Normal Subgroups, Group Homomorphisms
- § 11.1, 11.2 [Justin Chen]: Group Homomorphisms, The Isomorphism Theorems

- Week 7 (Feb. 29, March 2, 4):

**Rings**

- Week 8 (March 7, 9, 11):
- § 16.1: Rings
- § 16.1, 16.2: Rings, Integral Domains and Fields
- § 16.2, 16.3: Integral Domains and Fields, Ring Homomorphisms and Ideals

- Week 9 (March 14, 16, 18):
- § 16.3, 16.4: Ring Homomorphisms and Ideals, Maximal and Prime Ideals
- § 16.4: Maximal and Prime Ideals
- § 16.5: An Application to Software Design

- Week 10 (March 28, 30, April 1):
- § 17.1: Polynomial Rings
- § 17.2: The Division Algorithm
- § 17.2, 17.3: The Division Algorithm, Irreducible Polynomials

- Week 11 (April 4, 6, 8):
- Week 12 (April 11, 13):
- § 18.1, Macaulay2 Tutorial: Fields of Fractions, M2 Example, M2 Output from Example. See Software for instructions on using M2 via Sage Math Cloud and for download links etc.
- § 20.1, 20.2, 20.3: Vector Spaces

**Fields**

- Week 12 (April 15):
- § 20.3, 21.1: Vector Spaces, Extension Fields

- Week 13 (April 18, 20, 22):
- § 21.1: Extension Fields
- § 21.1: Extension Fields
- § 21.1: Extension Fields, Algebraic Closure

- Week 14 (April 25, 27, 29):
- § 21.1: Extension Fields, Algebraic Closure (Notes were continued from April 22, same file for both days)
- § 21.2, 22.1: Splitting Fields, Structure of a Finite Field
- § 22.1: Structure of a Finite Field

**Exam Review**

- RRR Week (May 2, 4):
- Review Session one on Fields, Chapters 21 and 22
- Review Session two (notes were continued inline, and hence are the same for both days)

### Assignments:

Note that only the questions marked "Hand in" should be handed in, those marked "Practice Problems" are not to be handed in, but may be helpful preparation for quizzes and exams. Of the "Hand in" questions 2-3 will be marked (most weeks).- Assignment 1. [Submitted Wednesday, January 27].
**Solutions to problems**:- Hand in:
- § 1.3 of Judson: 15, 19, 29
- § 2.3 of Judson: 12, 18, 28

- Practice Problems:
- § 1.3 of Judson: 10, 14, 18
- § 2.3 of Judson: 1, 3, 14, 23, 25, 27

- Hand in:
- Assignment 2. [Submitted Wednesday, February 3].
**Solutions to problems**:- Hand in:
- § 3.4 of Judson: 2, 7, 12, 34, 41, 54

- Practice Problems:
- § 3.4 of Judson: 15, 16, 25, 31, 39, 46, 49
- § 4.4 of Judson: 1, 2

- Hand in:
- Assignment 3. [Submitted Wednesday, February 10].
**Solutions to problems**:- Hand in:
- § 4.4 of Judson: 39, 44
- § 5.3 of Judson: 30, 33

- Practice Problems:
- § 4.4 of Judson: 28, 31, 34, 38
- § 5.3 of Judson: 21, 23

- Hand in:
- Assignment 4. [Submitted Wednesday, February 17].
**Solutions to problems**:- Hand in:
- § 5.3 of Judson: 29
- § 6.4 of Judson: 6, 8, 9, 16, 21

- Practice Problems :
- § 5.3 of Judson: 36
- § 6.4 of Judson: 3, 4, 19, 22, 23

- Hand in:
- Assignment 5. [Submitted Wednesday, February 24].
**Solutions to problems**- Hand in:
- § 9.3 of Judson: 9, 14, 34, 41, 46

- Practice Problems :
- § 9.3 of Judson: 4, 8, 13, 42

- Hand in:
- Assignment 6:
- There is no hand-in portion to this assignment due to there being a Midterm on Friday March 4.
- Practice Problems
**Solutions to selected problems**:- § 9.3 of Judson: 22, 23, 24, 25
- § 10.3 of Judson: 4, 5, 6, 7, 9, 12, 14
- § 11.3 of Judson: 12, 13, 14, 15, 16, 17, 18

- Assignment 7. Submitted Wednesday, March 16.
**Solutions to selected problems**- Hand in:
- § 16.6 of Judson: 12, 18, 29, 30, 31, 34

- Practice Problems:
- § 16.6 of Judson: 2, 8, 9, 11, 21, 22, 23, 26, 28

- Hand in:
- Assignment 8. Submitted Wednesday, March 30.
**Solutions to selected problems**- Hand in:
- § 16.6 of Judson: 27, 37, 38, 40
- Note that in #40 the book uses the notation $\mathrm{x=r\; (mod\; I)}$ , this is equivalent to saying that $\mathrm{x+I=r\; +I}$ or that $\mathrm{x=r\in \; R/I}$ , or that there exists some $\mathrm{s\in \; R}$ , $\mathrm{t\; \in \; I}$ such that $\mathrm{x=r\; +st}$ . A familiar example of this is that when we write $\mathrm{x=r\; (mod\; n)}$ for $\mathrm{x,r,\; n\; \in}\mathbb{Z}$ this is the same as writing $\mathrm{x=r\; (mod\; n}\mathbb{Z})$.

- § 16.6 of Judson: 27, 37, 38, 40
- Practice Problems :
- § 16.6 of Judson: 4, 7, 10, 24, 25

- Hand in:
- Assignment 9. Submitted Wednesday, April 6:
**Solutions to selected problems**- Hand in:
- § 17.4 of Judson: 18, 20, 26, 27

- Practice Problems:
- § 17.4 of Judson: 3, 7, 8, 11, 12, 13, 14, 15, 17, 21, 24, 25, 28, 29

- Hand in:
- Assignment 10.
- Practice Problems:
- § 18.3 of Judson: 3, 4, 5, 6, 7, 8, 9, 10, 11

- There is no hand in portion of this assignment, however there is a quiz on Friday April 15 covering the material from 18.1 of the text.

- Practice Problems:
- Assignment 11. Submitted Wednesday April 20.
- Hand in:
- § 20.4 of Judson: 16, 17, 18.
**Hint/explanation for 18 (d).**

- § 20.4 of Judson: 16, 17, 18.
- Practice Problems:
- § 20.4 of Judson: 3, 4, 5, 6, 7

- Hand in:
- Assignment 12. Submitted Friday April 29.
**Solutions to selected problems**- Hand in:
- § 21.4 of Judson: 18, 21, 22, 25
- § 22.3 of Judson: 3, 4, 15

- Practice Problems: Solutions to selected problems
- § 21.4 of Judson: 1, 2, 3, 4, 11, 12, 13, 14, 15, 16, 17, 23, 26, 27
- § 22.3 of Judson: 1, 2, 6, 8, 12, 14, 17, 18, 19, 20, 21, 22, 23

- Hand in:

### Algebra Software:

Macaulay2 (M2 for short) and Sage are both excellent open source computer algebra systems with some very helpful functions for algebra, algebraic geometry and number theory (among other things).- Macaulay2
- Download and install
- Use online
- M2 may also be used online via the Sage Math Cloud. To use M2 this way create an account (this is free) and open a new project, then start a new Terminal (>_ icon). Once the Sage terminal opens type the command: M2. M2 will then open and you can enter M2 commands.
- The M2 example from lecture: M2 Example, M2 Output from Example
- Computations in polynomial rings:
- Polynomial Rings
- Quotient Rings
- Factoring Polynomials
- Computing gcd
- Division with remainder, i.e. getting a representative in a quotient ring, inverting an element in a quotient ring which is also a field etc.
- Check if an integer or ideal is prime
- Check equality of ideals (or numbers or other objects)
- Sage
- Sage Cell
- Sage Math Cloud
- Computations in polynomial rings:

**Bonus Assignment using Macaulay2 or Sage**

- Due Friday April 29, in class. You may hand it in sooner if you like. Bonus Assignment
- Submission Format: You may just submit the text of your M2 or Sage session including the commands and output, i.e. something like the output from the example we did in class. Either print and hand in in class or submit a plain text file by email. Please include your name as a comment in the file.

### Quiz Dates and Solutions:

- Quiz 1 [Friday February 12]: Quiz 1 Solutions
- Quiz 2: [Friday March 18]: Quiz 2 Solutions
- Quiz 3: [Friday April 15]: Quiz 3 Solutions
- Quiz 4: [Friday April 22]: Quiz 4 Solutions:

### Homework Policy:

Homework will be due once a week (most weeks) usually on Wednesdays at the beginning of class, as a rule late homework will not be accepted. Homework should be handed in on paper in class (this is simpler for the grader). If you are not able to attend class on a given day alternate submission arrangements are possible (such as via email); assignments may also be slid under my office door. Paper submission is preferred, however if you do submit by email please submit .pdf files only (scans are fine). Homework due dates will be posted on this website along with the assignments. Homework assignments will be posted above at least a week before they are due. Each problem set will have a few problems (usually 5-6) that will be handed in, of these 2-3 will be graded. There will also be a longer list of practice problems. The material on the practice problems will be covered on quizzes and exams.Some things to keep in mind when doing your homework:

- You are encouraged to discuss problems with your classmates and are free to consult online resources. Working together on math problems can be an excellent way to learn and the internet is a useful resource.
**However**your final written solutions you hand in must be**your own work**written in**your own words**, that is your final solutions must be written by yourself without consulting someone else's solution. - All solutions should be written in
**complete, grammatically correct, English**(or at least a very close approximation of this) with mathematical symbols and equations interspersed as appropriate. These solutions should carefully explain the logic of your approach. - All proofs must be
**complete and detailed**for full marks. Avoid the use of phrases such as 'it is easy to see' or 'the rest is straightforward', you will likely be docked marks. Proofs in your homework should be**clear and explicit and should be more detailed than textbook proofs**. - If the grader is unable to make out your writing then this may hurt your mark.

### Grading:

Your grades will be broken down as follows:

- Assignments: 20%
- Quizzes: 10%
- Midterms: 30%
- Final Exam: 40%

**two**quiz marks not counting toward your grade. The quizzes will be short (~15 minutes) and will be done at the end of class. Additionally your

**one**lowest homework assignment mark will not count toward your final grade.

### Exam Dates:

- Midterm Exam 1: Friday March 4. 3-4 PM (i.e. in class). 9 Evans Hall.
- Midterm Exam 2: Friday April 8. 3-4 PM (i.e. in class). 9 Evans Hall.
- Final Exam: Wednesday, May 11, 2016. 7:00-10:00 PM.
**458 Evans Hall**. - For the Final Exam you may prepare 5 pages double sided or 10 pages single sided of notes to use during the exam:
- A copy/scan of the notes must be submitted to me via email or on bCourses (or on paper as a last resort) on or before
**Friday May 6**. - The purpose of the notes is to allow you to write down important theorems, lemmas, etc.
- On the day of the exam please remember to
**bring your notes with you** - Please arrive at the exam room a couple minutes in advance so I can quickly glance over your notes that you bring to the exam to ensure they are a quite close match to the copy submitted online. Small additions/changes are okay, major additions (i.e. an additional 5 pages over the submitted size) are not.
- Notes may be handwritten or type written, your choice.