Introduction to Abstract Algebra (Math 113) - Summer 2014
University of California, Berkeley



MTWTh 4-5pm 71 Evans Hall (Lecture)
MTWTh 5-6pm 71 Evans Hall (Discussion Section)

Instructor: George Melvin

This summer there are two Math 113 courses taking place at Berkeley. It is a great pleasure to be teaching one of these classes along side my friend James McIvor - James is teaching the 10am-12pm Math 113 course. James and I want to provide a comfortable learning environment and to maximise the resources and opportunities available to students. As such, we are going to be teaching 'in tandem' - this means that both courses will have the same homework assignments and worksheets and will be taught the same material. The aim is to allow students from each class to discuss the material with each other without disconnect. Also, there will be a shared piazza.com forum and students will have the opportunity to attend the office hours of either instructor.

However, you must attend the lecture/discussion section for which you are registered.

Contact

Email:
Office: 853 Evans Hall
Office Hours: Tuesday 2-4pm, Wednesday 2-4pm. All office hours will take place in 853 Evans Hall.

For James's office hours see his course page here.

Overview

The aim of this class is to introduce the student to more powerful methods of abstraction and highlight similar structures that appear in different mathematical settings: for example, the rational numbers and real numbers possess a similar structure in that every nonzero number admits a multiplicative inverse, while that same is not true of the integers. However, all three of these examples of number systems possess a similar structure - there are notions of addition and multiplication. This should lead the budding mathematician to wonder whether there exist other mathematical settings where we can 'add' and 'multiply' - the set of 2x2 matrices with real entries provides a nontrivial example - and so our investigations take shape (this leads to the study of rings).

There will be a strong emphasis on examples throughout the course and 'getting your hands dirty' by doing lots of computations. Many examples will be introduced and will appear frequently as the semester progresses so that it is important for the student to feel comfortable with playing around with these examples. This will require frequent class attendance and asking lots of questions!

Ultimately, I hope that the course will be a fun, challenging and rewarding experience for everyone involved :)

Course Outline

Content: From the online schedule: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

Prerequisites: Math 54, although some exposure to writing proofs will be helpful. Above all, students will be expected to have that oft sought after quality mathematical maturity (funnily, not the same as social maturity).

Grading: There is a total of 400 385 points available:

• Homework (120 points): Eight assignments, each worth 15 points. Homework (105 points): Seven assignments, each worth 15 points.
• Exam 1 (125 points)
• Exam 2 (125 points)
• Participation (30 points)

If, due to unforeseen circumstances (medical emergency etc.), you are unable to sit one of the exams then the remaining exam will be worth 178 points, and the homework will be worth a total of 192 points (eight assignments, each worth 24 points). If you can't attend an exam then you must let me know as soon as possible; medical emergencies must be supported with a doctor's/nurse's note.

Textbooks: It is important to realise that there is no *required* textbook for this course (despite what appears on the Online Schedule). James and I will be providing thorough and complete notes that will contain all of the material you are required to know for exams, homeworks etc. These notes will be the 'course textbook'. I shall be writing the notes for the first half of the course on Group Theory and James will write the notes for the second half of the course on Ring Theory. Group Theory notes will be distributed at the beginning of the course and James's notes will be distributed a few weeks into the semester. All notes will be posted below.

The 'required' textbook for this course (as designated in the online schedule) is 'A First Course in Abstract Algebra' by Fraleigh. This is not our choice of textbook and it is very expensive ($130); let me clarify - the textbook is a balanced and readable introduction to the subject but the course will not follow the structure of this textbook 'verbatim'; in particular, there will be a stronger emphasis on group actions in the group theory part of the course. Furthermore, I feel that it is outrageous to pay such a huge sum of money for this book. As such, it has been demoted to the status of 'recommended' textbook; if you have already purchased a copy of this book then it will be of use as a supplementary reference to the Course Notes provided, but you may be interested in purchasing one of the (cheaper) textbooks below.

Here are some other 'recommmended' textbooks that are better (in my and James's opinions) and cheaper:

• Topics in Algbera - Herstein (Second Edition; $20 here) Written an an older style but many *great* problems. This was the initial choice for the textbook for this course (ie the choice that James and I would make).
• Basic Algebra I - Jacobson (Second Edition, Dover Publications; $15) Quite terse but very well written. Cheap!
• A Book of Abstract Algebra - Pinter (Second Edition, Dover Publications; $15) Recommended by a former student for its clarity and insight to abstract concepts. This is a very accessible textbook and available at amazon.com. If you feel it necessary to have a textbook for this course, it would be a good choice.
• Abstract Algebra - Dummit & Foote (Third Edition; $75) Widely used in US Math Departments. Pretty verbose and contains much more material than we'll cover; it's a thick book. Lots of problems, most solutions available online.
• Algebra - Artin (Second Edition; $130) This is the book that is frequently used at MIT, where Artin is a Professor. It is a great (but hard) introduction to algebra at the undergraduate level.
• Classic Algebra - Cohn (Recently revised; $75) This is a fantastic introduction to algebra but it is quite a challenging book; it has been used at the University of Cambridge as a recommended textbook for their algebra courses.
• Group Theory - Milne (Available online) These notes are very well-written and provide a broad introduction to group theory, although they are perhaps a bit sophisticated for this course. They might be useful for those of you wishing to stretch yourself as there are many great exercises.

If you are considering taking further algebra classes (eg Math 250A/B at Berkeley) or considering going to graduate school to study mathematics/physics (and also chemistry/biology) then the following textbooks are introductions to algebra at a graduate (= 'mature') level - they are *hard*:

• Algebra - Lang This is a classic reference book and contains far more information than you are ever likely to require. The exercises are *hard* and the material is discussed at a high level of abstraction. This is the textbook frequently used for the Math 250A course at Berkeley (and also the firstalgebra book that I bought as an undergraduate!).
• Algebra - Hungerford A good compliment to Lang's Algebra.
• An Introduction to Commutative Algebra - Atiyah & MacDonald This (little) book only focuses on commutative rings (these objects will appear in the latter half of the course) and is possibly one of the finest books on mathematics ever written! Great exercises, excellent clarity and presentation, just a wonderful book. If you are looking to go to graduate school to study mathematics (with an algebraic emphasis) or theoretical physics (string theory etc) then this book will become your friend/nemesis (it is the second algebra book that I bought as an undergraduate!).

Most (hopefully all) of the above textbooks will be placed on 'reserve' in the Mathematics Library in Evans Hall - you can also ask to look at a personal copy owned by either James or myself (if this is the case!). I recommend having a look at some of these texts if you get the opportunity but remember that they are considered supplementary to the Course Notes that will be distributed. Also, be aware that utilising too many references can often be a hinderance - try to find a single book that you enjoy and stick with it. Feel free to ask any questions you have concerning the suitability of a textbook.

If you find some notes online (there are *lots* of people who have written 'Introduction to Abstract Algebra' notes!) and find them helpful then let either myself or James know and we will provide a link to them above.

ADDED 8/5: Here are some excellent resources on Galois theory:

• Galois Theory - E. Artin *THE* modern treatment of Galois theory taken from a series of lectures by Emil Artin, the mathematician who provided us with our treatment of the subject today. Available at Amazon.com for $6!
• Lecture notes on Galois theory - A. Baker Available here - we will be covering Ch. 1-4
• Galois Theory - Pavaman Murthy et al. Available here, and contains all the group theory and linear algebra required, in addition to the main results we will cover.
• Galois Theory - M. Reid Notes from a course at Warwick University by Prof. Miles Reid, an excellent expositor of undergraduate (andy beyond!) algebra. Provides some background to the theory of equations.

Progress

It is important to realise that this course will be fast-paced and, due to the nature of a Summer Sessions course, you will be introduced to a lot of new material in a short period of time. It is essential that you attend class.

Here is what we actually covered in class:

• 6/23 - Symmetries of the square. Definition of a group. Wiring diagrams, W_3. Symmetries of tetrahedron. f:T-->W_4
• 6/24 - Fundamental theorem of arithmetic, division algorithm, gcd, Euclid algorithm. Equivalence relation, equivalence class. Partition of a set into equivalence classes. Equivalence classes are equal or disjoint. Z/nZ.
• 6/25 - Definition of group, subgroup. Abelian, finite, infinite groups. Group generated by g, . is either infinite or g^{k}=e. Order of element o(g). Examples.
• 6/26 - Homomorphisms, kernel, image. Examples. Isomorphisms. Well-definedness of functions with domain Z/nZ.
• 6/30 - Cosets. Left H-partition. Lagrange's Theorem. Order of element divides order of group. Groups of prime order are cyclic. Cyclic groups, generators. Statement of structure theorem of cyclic groups.
• 7/1 - (Left) group actions of G on S. Group actions of G on S are equivalent to homomorphisms G-->perm(S). Examples. Orbits, stabilisers. Fundamental group actions: G acting by conjugation on G, permutation action of G on G, G acting by conjugation on Sub(G). Orbits are equivalence classes. Orbit decomposition formula.
• 7/2 - Normalisers, centralisers. Conjugacy classes. Examples. Beginning to discuss cycles in S_{n}.
• 7/3 - More about cycles in S_{n}. Classification of conjugacy classes in S_{n}, partitions of integers. Cycle type of a permutation. Computations. Statement of Orbit-Stabiliser Theorem.
• 7/7 - Proof of Orbit-Stabiliser Theorem. p-groups. Fixed point congruence for p-group actions. Existence of elements of order p.
• 7/8 - Sylow Theorems, statements and proofs. Groups of order pq are not simple.
• 7/9 - Quotient groups, examples. Quotient homomorphism. First Isomorphism Theorem.
• 7/10 - Direct products. Structure Theorem of finite abelian groups.
• 7/14 - Statement of classification of finite abelian p-groups. Elementary divisors, invariant factors. Classification of finite abelian groups. List of isomorphism classes of finite abelian groups of order 352.
• 7/21 - Definition of rings. Basic examples. Zerodivisors, nilpotent elements, units, idempotents.
• 7/22 - Subrings, R[A] (adjoining (sets of) elements). Product rings. Homomorphisms, isomorphisms. Trivial kernel = injective. Examples.
• 7/23 - Kernels, ideals. Examples. (X) - ideals generated by X. Intersections, sums of ideals.
• 7/24 - Review of cosets in abelian groups. Construction of quotient ring - well-definedness. Quotient homomorphism. First Isomorphism Theorem for rings. Examples, computations.
• 7/28 - Factorisation of integers/polynomials; irreducible polynomials. Eisenstein criterion. (James covered this lecture)
• 7/29 Fields, domains, Euclidean domains, principal ideal domains, unique factorisation domains. Examples. The 'containment' diagram.
• 7/30 - Proof of containment diagram: field --> Euclidean domain --> PID --> UFD --> domain. Examples and non-examples of each class of ring. Definition of prime ideals, maximal ideals. Maximal implies prime (but not conversely!).
• 7/31 Examples of prime, maximal ideals; how to prove primality, maximality. Prime <--> R/I domain. Maximal <--> R/I maximal. Applications.
• 8/4 Field extensions. Examples. Finite/infinite extensions. Algebraic/transcendental elements, algebraic/transcendental extensions. Simple extensions. Classification of simple extensions. Examples.
• 8/5 Proof of classification of simple extensions. Definition of minimal polynomial. Minimal polynomial divides any other polynomial relation. Finite implies algebraic. Computations and examples.

Lecture Notes

All lecture notes will be posted here. If you find any mistakes (mathematical and/or grammatical) then please let me know.

Here are some notes discussing set notation, basic set theoretic operations and functions. It also includes some of the common notation we will encounter.

• Group Theory (Version 4) (July 8, 2014) Lecture 7 has been updated - there was a mistake in the definition of a k-cycle, and in the proof of the classification of conjugacy classes. Lecture 9 was updated - Corollary was added.
• Ring Theory (Updated, 8/4)

Exams

All exams will take place in 71 Evans during class.

Exam 1 When: 17 July 2014, 4-6pm, Material: Group Theory

Exam 2 When: 14 August 2014, 4-6pm, Material: Ring Theory

Practice Exam 1 Solution

Practice Exam 2 Solution

Exam 1 Solution (George's Class)

Exam 2 Solution (George's Class)

Here is a Math 113 past exam archive.

Important: Students with DSP requirements must let me know as soon as possible (ie, within first week of course) and provide all necessary documentation. Feel free to send me an email or speak to me after class about any issues.

Resources

piazza.com There is a class forum at piazza.com, if you would like to added to this then please send me an email. Use this forum to ask any questions you have concerning the material covered during class and on any problems you have with homework; also, feel free to answer any questions you feel comfortable discussing with your fellow students. Try to be civil with each other!

Notes: I will post any supplementary notes I prepare and hand out during the semester here.

• Syllabus
• Factsheet for the symmetric group.
• Worksheet to help you write basic proofs.

Extras: Here is some useful information on problem solving techniques given by the Hungarian mathematician George Polya.

Here are some useful notes written by Prof. George Bergman on basic mathematical language.

Here is some useful advice on how not to lose marks on exams by a former colleague Andrew Critch.

Here is an interesting article by Bjorn Poonen (a former mathematics professor at Cal, now at MIT) on why rings should contain 1.

Here is a discussion of the 'inverse Galois problem' - given a finite group G, does there exist a normal extension L of Q such that Gal(L,Q)=G?.

Homework

Homework is due each Thursday in class at 4.20pm, except for those weeks when exams are taking place, in which case homework will be due Wednesday in class at 4.20pm.

Late homework will not be accepted.

If you are unable to submit your homework at the required time then you can leave it outside my office (853 Evans Hall) at any time before it is due. Please email me if you intend to leave your homework outside my office.

Collaboration on homework is welcome and encouraged although if you are working with another student please state that you have done so (eg. if you work with E. Nother on a particular question just write "This question was completed with E. Nother."). However, all homework assignments must be written up individually. Failure to declare collaboration with another student will result in a grade penalty (and it is remarkably simple to tell when students have copied each other). Also, if you have used a textbook or online notes to help you understand/solve a problem please cite a reference (eg. if you used pages 52-60 of Prof. X's online lecture notes just write "This question used p.52-60 of Prof. X's online lecture notes, available at www.math.com/~profx/linalg)

Grading: Each week three problems will be selected at random to be graded. If you have a query concerning the grading of your homework then please contact me.

Homework Assignments:

• Homework 1 Due Thursday June 26 Solution and supplement on symmetries of tetrahedron.
• Homework 2 Updated 7/1 Due Thursday July 3 Solution
• Homework 3 Due Thursday July 10 Due Monday July 14 Solution
• Homework 4 Due Wednesday July 16 Solution
• Homework 5 Due Monday July 28 Solution
• Homework 6 Due Monday August 4 (Updated 8/1 - Problem 4 hint modified to include 'nonzero') Solution (Solution updated 8/5)
• Homework 7 Due Wednesday August 13

Worksheets

Here are the worksheets that are handed out during discussion section. You should use these worksheets to get extra practice at computations. They will also highlight various consequences of Theorems you will see in this course. If you have any questions on the worksheets then please get in contact with me; better still, ask a question at piazza.com (making sure to remember to state which problem you are working on!)

• June 24 Solution
• June 25 Solution
• June 26 (Note: in Q1c) the codomain should be Z/7Z) Solution
• June 30 Solution
• July 1 Solution
• July 2 Solution
• July 3 Solution
• July 7 Solution
• July 8 Note that this is a different version from that distributed in class - an erroneous problem has been removed Solution
• July 9 Solution
• July 14 Solution
• July 21 Solution
• July 22 Solution
• July 23 Solution
• July 24 Solution Supplement to solution
• July 28 Solution
• July 30 Solution
• August 4 Solution
• August 6 Solution
• August 11 Solution