Math 113: Abstract algebra
UC Berkeley, Fall 2010
Section 2, T/Th
4, T/Th 3:40-5
Office hours are cancelled Wednesday, November 24
Announcement: There will be review on Thursday, December 9, same time
& place as class.
ianagol at math.berkeley.edu
Office phone: 510-642-4377.
Office: 921 Evans.
Office Hours: Monday 11:10-12 am, Wednesday 2:10-4 pm, or by call or
e-mail to set an appointment
In previous courses you have seen many kinds of algebra, from the
algebra of real and complex numbers, to polynomials, functions,
vectors, and matrices. Abstract algebra (most mathematicians would
just call this "algebra", I'm not sure why the word "abstract" is
there) encompasses all of this and much more. Roughly speaking,
abstract algebra studies the structure of sets with operations on
them. We will study three basic kinds of "sets with operations on
them", called Groups, Rings, and Fields.
A group is, roughly, a set
with one "binary operation" on it
satisfying certain axioms which we will learn about. Examples of
groups include the integers with the operation of addition, the
nonzero real numbers with the operation of multiplication, and the
invertible n by n matrices with the operation of matrix
multiplication. But groups arise in many other diverse ways. For
example, the symmetries of an object in space naturally comprise a
group. After studying many examples of groups, we will
develop some general theory which concerns the basic principles
underlying all groups.
A ring is, roughly, a set
with two binary operations on it
satisfying certain properties which we will learn about. An example
is the integers with the operations of addition and multiplication.
Another example is the ring of polynomials. A field is a ring with
certain additional nice properties, such as the rational and real
numbers. At the end of the course we will
have built up enough machinery to prove that one cannot trisect a
sixty degree angle using a ruler and compass.
In addition to the specific topics we will study, which lie at the
foundations of much of higher mathematics, an important goal of the
course is to expand facility with mathematical reasoning and proofs in
general, as a transition to more advanced mathematics courses, and for
logical thinking outside of mathematics as well. I am hoping that
you already have some familiarity with proofs from Math 55 or a
similar course. If not, the following might be helpful:
Some notes of Michael Hutchings giving
a very basic introduction to proofs are available here.
for writing mathematics.
The textbook for this course is John B. Fraleigh,
A first course
in abstract algebra , 7th edition, Addison-Wesley.
This book is very readable, has been well liked by students in the
past, and contains lots of good exercises and examples.
Most of the lectures will correspond to particular sections of the
book (indicated in the syllabus below), and studying these sections
should be very helpful for understanding the material. However,
please note that in class I will often present material in a different
order or from a different perspective than that of the book. We will
also occasionally discuss topics which are not in the book at all.
Thus it is important to attend class and, since you shouldn't
to understand everything right away, to take good notes.
There are many other algebra texts out there, and you might try
browsing through these for some additional perspectives. (Bear in
mind that Fraleigh is an "entry-level" text, so many other algebra
books will be too hard at this point; but after this course you should
be prepared to start exploring these. There is a vast world of
algebra out there!)
In addition, the math articles on wikipedia
have gotten a lot better
than they used to be, and much useful information related to this
course can be found there. However you shouldn't blindly trust
anything you read on the internet, and keep in mind that wikipedia
articles tend to give brief summaries rather than the
detailed explanations that are needed for proper understanding.
Abstract and Concrete, Edition 2.5, Frederick M. Goodman
Abstract Algebra, Rudolf Lidl and Gunter Pilz: this book
gives applications of abstract algebra, but is a second course (you can
read online through the library).
Algebra, Groups, Rings and Fields, P. M. Cohn
Algebra, Paul Garrett (chapter 01 has some background on number
Homework is due every Thursday (except for the first two Thursdays
and the weeks of the midterms or Thursday holidays) at the beginning of
class. 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!) Homework
assignments will be posted below at least a week before they are due.
No late homeworks will be accepted for any reason, so that we can go
over the homework problems at the beginning of Thursday's class
(which is when people are most eager to see solutions to troublesome
problems). However it is OK if you miss the deadline once or twice,
because your lowest two homework
scores will be dropped.
- It is essential to thoroughly learn the definitions of
concepts we will be studying. You don't have to memorize the exact
wording given in class or in the book, but you do need to remember all
the little clauses and conditions. If you don't know exactly what a
UFD is, then you have no hope of proving that something is or is not a
UFD. In addition, learning a definition means not just being able to
recite the definition from memory, but also having an intuitive idea
of what the definition means, knowing some examples and non-examples,
and having some practical skill in working with the definition in
- In the same way it is necessary to learn the statements of
the theorems that we will be proving.
- It is not necessary to memorize the proofs of theorems. However
the more proofs you understand, the better your command of the material
will be. When you study a proof, a useful aid to memory and
understanding is to try to summarize the key ideas of the proof in
a sentence or two. If you can't do this, then you probably don't
yet really understand the proof.
- The material in this course is cumulative and gets somewhat
harder as it goes along, so it is essential that you do not fall
- If you want to really understand the material, the key is to ask
your own questions. Can I find a good example of this? Is that
hypothesis in that theorem really necessary? What happens if I drop it?
Can I find a different proof using this other strategy? Does that other
theorem have a generalization to the noncommutative case? Does this
property imply that property, and if not, can I find a counterexample?
Why is that condition in that definition there? What if I change it
this way? This reminds me of something I saw in linear algebra; is
there a direct connection?
- If you get stuck on any of the above, you are welcome to come
to my office hours. I am happy to discuss this stuff with you.
Usually, the more thought you have put in beforehand, the more
productive the discussion is likely to be.
When preparing your homework, please keep the following in mind:
1) You are encouraged to discuss the homework problems with your
classmates. Mathematics can be a fun social activity! Perhaps 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 time to write down your
solutions to hand in, you must do this by yourself, in your own
words, without copying or looking at someone else's paper. If you obtain help from another student
with a problem, write this down on your homework. This will not
affect your grade, unless it is clear that you have copied verbatim -
the final answer should be written in your own words. It is important
to get in the habit of citing your sources, which may include your
colleagues (otherwise, you are plagiarizing)!
2) All answers should be written in complete, grammatically
English sentences which explain the logic of what you are doing,
with mathematical symbols and equations interspersed as appropriate.
For example, instead of writing "x^2 = 4, x = 2, x = -2", write "since
x^2 = 4, it follows that x = 2 or x = -2." Otherwise your proof
be unreadable and will not receive credit. Results of calculations
and answers to true/false questions etc. should always be justified.
Proofs should be complete and detailed. The proofs in the book
provide good models; but when in doubt, explain more details. Avoid
phrases such as "it is easy to see that"; often what follows such a
phrase is actually a tricky point that needs justifiction, or even
false. You can of course cite theorems that we have already proved in
class or from the book.
Reading Assignments (from
- 8/31: Read Notes on
for writing mathematics, and Sections 0, 1 of Fraleigh
- 9/2: Read Sections 2 and 3
- 9/7: Read Sections 3 and 4
- 9/9: Read Sections 5 and 6
- 9/14: Read Sections 6 and 7
- 9/16: Read Sections 8 and 9
- 9/21: Read Sections 8 and 9
- 9/23: Read Sections 9 and 10
- 9/28: Read Sections 10 and 11
- 9/30: Read Sections 11 and 12
- 10/5: Read Section 12
- 10/7: Midterm 1
- 10/12: Read Sections 12 and 13
- 10/14: Read Section 13
- 10/19: Read Section 14
- 10/21: Read Section 15
- 10/26: Read Section 18
- 10/28: Read Section 19
- 11/2: Review
- 11/4: Midterm 2
- 11/9: Read Section 20
- 11/11: Veteran's day, no class
- 11/16: Read Section 21
- 11/18: Read Section 22
- 11/23: Read Section 23
- 11/25: Thanksgiving, no class
- 11/30: Read Section 26
- 12/2: Read Section 27
- Homework 1 due 9/9 (at the beginning of class): Section 0,
#29-34*; Section 2, #5,6,9,13,23,37, *explain for each problem from
Section 0 which property of equivalence relations hold, and which do
not, with justification. Homework 1 solutions (cut and paste into
a web browser):
- Homework 2 due 9/16: Section 3 # 2, 4, 6, 27, 33 ;
Section 4 # 9, 10, 37; Section 5 # 13, 23, 49; Section 6 # 22, 48.
Homework 2 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_2_sol.pdf
- Homework 3 due 9/23: Section 7 # 11, 18, Section 8 # 3,
8, 21, 45, 46. Homework 3 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_3_sol.pdf
- Homweork 4 due 9/30: Section 9 # 7, 13, 16, 29, Section
10, # 6, 7, 28, 29, 39. Homework 4 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_4_sol.pdf
to Midterm 1
There will be in-class midterms on 10/7 and 11/4, and a final exam on
12/15, 8-11 am, 6 Evans (Section 2) or
12/17, 7-10 pm, 71 Evans (Section 4).
- Homework 5 due 10/14: Section 11, # 10, 18, 20, 30, 47,
Section 12 #8, 16, 24-30, 38, 40. Homework 5 solutions:
- Homework 6 due 10/21: Section 13, # 2, 6, 8, 20, 36, 37, 44,
50 Section 14 # 3, 6, 27, 31, 32, 33. Homework 6 solutions:
- Homework 7 due 10/28: Section 15, # 3, 7, 13, 36, 37,
Section 18 # 9, 15, 20, 25, 37, 41. Homework 7 solutions:
- No homework due 11/11 because of Veteran's day
- Solutions to Midterm 2 Section 2:
- Solutions to Midterm 2 Section 4:
- Homework 8 due 11/18: Section 19, #3, 9, 25, 29;
Section 20 #6, 7, 14, 27, 28; Section 21 #1, 2, 5, 7. Homework 8
- Homework 9 due 12/2: Section 22, # 5, 14, 25, 30; Section 23
#3, 12, 17, 28, 34; Section 26 #3, 17, 18, 20; Homework 9 solutions:
There will be no makeup exams. However you can miss
one midterm without penalty, as explained in the grading policy below.
There is no regrading unless there is an egregious error
such as adding up the points incorrectly. Every effort is made to
grade all exams according to the same standards, so regrading one
student's exam would be unfair to everyone else.
grade will be determined as follows: homework 20%, midterms 20% each,
final 60%, lowest exam score -20%. All grades will be curved to a
uniform scale before being averaged.
- Preliminaries. We will begin with a review of some
essential preliminaries, including sets, functions, relations,
induction, and some very basic number theory. You have probably already
seen this material in Math 55 or elsewhere, so the review will be
brief. Some of this material is in section 0 of the book, some
is scattered throughout random later sections, some is in the above
notes on proofs, and some is in none of the above.
- Groups. We will learn a lot about groups, starting with
the detailed study of a slew of examples, and then proceeding to some
important general principles. We will cover most of Parts I, II, and
III of the book. We will consider a few examples which are not in the
book, such as symmetry groups of polyhedra and wallpaper groups. We
skip the advanced group theory in Part VII, aside from stating a
couple of the results. (You can learn some of this material in Math
114.) We will completely skip Part VIII on group theory in topology;
this material is best learned in a topology course such as Math 142.
- Ring theory and polynomials. Next we will learn about
rings. We will pay particular attention to rings of polynomials,
which are very important e.g. in algebraic geometry. We will cover
most of Parts IV, V, and IX.
- Elements of field theory. Finally, after reviewing some
notions from linear algebra in a more general setting, we will learn
the basics of fields, from Part VI of the book. We will develop
enough machinery to prove that one cannot trisect a sixty degree angle
with a ruler and compass. We will not have time for the more advanced
field theory in Part X, including the insolvability of the quintic;
this is covered in Math 114.
About the course. Brief review of preliminaries: sets, functions,
injections, surjections, bijections. Equivalence relations and modular
arithmetic. Proof by induction, strong induction, and the well ordering
principle. The division theorem. Greatest common divisors, the
Euclidean algorithm, and solving ax = b (mod n). The fundamental
theorem of arithmetic. [0,notes on proofs]. For fun: Play
the game Set
- 9/2: Binary
operations. Isomorphism of binary structures.
Using structural properties to show that binary structures are not
isomorphic. [Sections 2,3]
- 9/7: Groups: definition, many examples [Section 4]
- 9/9: Subgroups, cyclic groups [Sections 4,5]
- 9/14: cyclic groups [Section 6]
- 9/16: Cayley graphs [Sections 7]
- 9/21: permutation groups [Sections 8, 9]
- 9/23: orbits, alternating groups, cosets [9,10]
- 9/28: Lagrange's theorem 
- 9/30: Product groups 
- 10/5: review
- 10/12: plane isometries, frieze and wallpaper groups
- 10/14: homomorphisms 
- 10/19: factor groups 
- 10/21: factor group computations and simple groups 
- 10/26: rings and fields 
- 10/28: integral domains 
- 11/9: Fermat's and Euler's theorems 
- 11/16: Field of quotients 
- 11/18: Polynomial rings 
- 11/23: Factorization of polynomials 
- 11/30: Homomorphisms and factor rings 
- 12/2: Prime and maximal ideals