Math 113: Abstract Algebra

University of California, Berkeley. Spring 2017.

Tuesday/Thursday from 12:30-2:00 PM in 3105 Etcheverry 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 **Justin Chen**.

- GSI Office Hour Location: 732 Evans Hall
- GSI Office Hour Times:
- Monday, Tuesday, Thursday: 1 -- 3 PM
- Wednesday: 1 -- 5 PM

### Piazza Q&A :

There is a Piazza page for Q&A discussions for this course; click here to join.### Book:

Abstract Algebra: Theory and Applications by Judson (2016 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. 17, 19):

**Groups**

- Week 2 (Jan. 24, 26):
- § 3.3: Subgroups
- § 4.1, 4.2: Cyclic Subgroups, Multiplicative Group of Complex Numbers

- Week 3 (Jan. 31, Feb. 2 ):
- Week 4 (Feb. 7, 9):
- § 6.1, 6.2: Cosets, Lagrange's Theorem
- § 6.2, 6.3: Lagrange's Theorem, Fermat's and Euler's Theorems

- Week 5 (Feb. 14, 16):
- § 9.1: Isomorphisms (Definition and Examples)
- § 9.1, 9.2: Isomorphisms, Direct Products

- Week 6 (Feb. 21, 23):
- Week 7 (Feb. 28, March 2):
- Week 8 (March 7):
- § 11.2: The Isomorphism Theorems

**Rings**

- Week 8 (March 9):
- § 16.1, 16.2: Rings

- Week 9 (March 14, 16):
- § 16.2: Integral Domains and Fields
- § 16.3, 16.4: Ring Homomorphisms and Ideals

- Week 10 (March 21, 23):
- § 16.4, 17.1: Maximal and Prime Ideals
- § 17.1: Polynomial Rings

- Week 11 (April 4, 6):
- § 17.2: The Division Algorithm
- § 17.3: Irreducible Polynomials

**Fields**

- Week 12 (April 11, 13):
- § 18.1: Fields of Fractions. Macaulay2 Tutorial: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, 21.1: Vector Spaces, Extension Fields (Same file as April 11)

- Week 13 (April 18, 20):
- § 21.1: Extension Fields
- § 21.1: Extension Fields (Same File as April 18)

- Week 14 (April 25, 27):
- § 22.1, 21,2: Extension Fields, Splitting Fields
- § 22.1, 23.1: Structure of a Finite Field, Field Automorphisms (Same notes as April 25)

**Exam Review**

- RRR Week (May 2, 4):

### 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 Solutions to Marked Problems. Submitted Thursday January 26:
- Hand in:
- § 3.4: 4, 10, 14, 26, 32, 49

- Practice Problems:
- § 3.4: 2, 15, 16, 25, 31, 41, 51, 54

- Hand in:
- Assignment 2 Solutions to Marked Problems. Submitted Tuesday February 7:
- Hand in:
- § 4.4: 23, 24, 38, 39, 43
- § 5.3: 12, 13

- Practice Problems:
- § 4.4: 10, 11, 14, 26, 27, 28, 37, 44
- § 5.3: 6, 17, 27, 34

- Hand in:
- Assignment 3 Solutions to Marked Problems. Submitted Tuesday February 14:
- Hand in:
- § 5.3: 14
- § 6.4: 9, 11, 12, 16, 17

- Practice Problems:
- § 5.3: 16
- § 6.4: 2, 3, 4, 10, 13, 14, 18, 19

- Hand in:
- Assignment 4 Solutions to Marked Problems. Submitted Tuesday February 21:
- Hand in:
- § 6.4: 21
- § 9.3: 5, 9, 26, 28, 47

- Practice Problems:
- § 6.4: 22, 23
- § 9.3: 11, 18, 19, 20, 25, 27, 49

- Hand in:
- Additional Practice Problems for midterm:
- Practice Problems:
- § 9.3: 19, 21, 22, 23, 24, 31, 45, 48, 52
- § 10.3: 3, 4, 5, 6, 7, 8, 9, 11
- § 11.3: 2, 3, 4, 5, 6, 8, 9, 10, 11, 12

- Practice Problems:
- Assignment 5 Solutions to Marked Problems. Submitted Tuesday March 14:
- Hand in:
- § 11.3: 14, 16, 17, 18
- § 16.6: 1, 2

- Practice Problems:
- § 11.3: 13, 15, 19
- § 16.6: 3

- Hand in:
- Assignment 6 Solutions to Marked Problems. Submitted Tuesday March 21:
- Hand in:
- § 16.6: 9, 10, 18, 25, 27

- Practice Problems:
- § 16.6: 6, 7, 8, 11, 12, 16, 17, 20, 26

- Hand in:
- Assignment 7. Submitted Tuesday April 4:
- Hand in:
- § 16.6: 16, 21, 29, 38, 40
- § 17.4: 26
- Hint for §16.6 #29: First try to show that for all $\mathrm{a,b\; \in \; R}$ we have that $\mathrm{2ab=ab+ab=ba+ba=2ba}$ (using the notation for repeated addition) by considering expressions of the form ${\mathrm{(a+b)}}^{3}$, ${\mathrm{(a-b)}}^{3}$; you may also show that $\mathrm{6c\; =0}$ and that $\mathrm{3c+3{c}^{2}=0}$ for any element of $\mathrm{c\; \in \; R}$. Considering $\mathrm{c=a+b}$ would then be helpful.
- Note that in §16.6 #40 the book uses the notation $\mathrm{x=r\; (mod\; I)}$, this is equivalent to saying that $\mathrm{x+I=r\; +I}$ in $\mathrm{R/I}$.

- Practice Problems:
- § 16.6: 18, 19, 22, 26, 37

- Hand in:
- Assignment 8. Submitted Tuesday April 11:
- Hand in:
- § 17.4: 15, 16, 18, 24, 28, 29

- Practice Problems:
- § 17.4: 6, 7, 11, 12, 13, 14, 19, 20, 21, 25

- Hand in:
- Assignment 9 Solutions to Marked Problems. Submitted Thursday April 20:
- Hand in:
- § 18.3: 6, 8
- § 20.4: 1, 3
- § 21.4: 22

- Practice Problems:
- § 18.3: 7, 9, 10
- § 20.4: 2, 12, 14, 15
- § 21.4: 1, 2, 3, 4, 5, 9, 10, 12, 15, 16, 17, 18, 20 , 24, 26, 27

- Hand in:
- Assignment 10 Solutions to Marked Problems. Submitted Thursday April 27:
- Hand in:
- § 21.4: 2(a, g, h only), 11, 14, 23, 25
- § 22.3: 8, 13

- Practice Problems:
- § 21.4: 1, 2, 3, 4, 5, 9, 10, 12, 15, 16, 17, 18, 20 , 24, 26, 27
- § 22.3: 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

- Hand in:
- Bonus Assignment.
**Due Thursday May 4. Please submit electronically via bCourses**.

### 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:

### Quiz Dates and Solutions:

- Quiz 1: Thursday February 9 in class.
- Quiz 2: Thursday March 16 in class.
- Quiz 3: Tuesday April 4 in class.
- Quiz 4: Thursday April 20 in class. Solution to Quiz 4.

### Homework Policy:

Homework will be due once a week (most weeks) usually on Thursdays 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.

### Incomplete Grade Policy:

Per University policy an "incomplete" grade will be granted only in cases where a student has completed more than 75% of the term work, and is receiving a passing grade on this work, but is unable to complete the course due to documented circumstances beyond their control.### Grading:

Your grades will be broken down as follows:

- Assignments: 31%
- Quizzes: 12%
- Midterm: 20%
- Final Exam: 37%

**one**quiz mark 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: March 2 in class.
- Final Exam: Thursday May 11, 3:00 pm - 6:00 pm in 3105 Etcheverry Hall
- For the Final Exam you may prepare 5 pages double sided or 10 pages single sided of notes to use during the exam:
- 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** - Notes may be handwritten or type written, your choice.