Math 113: Abstract algebra
UC Berkeley, Fall 2009, room 4 Evans, MWF 2:10-3:00
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Wednesday 9:30 - 11:30.
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. The moves that one can do on Rubik's cube comprise a fun
example of 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. 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 old notes of mine giving
a very basic introduction to proofs are available here.
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 expect
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.
It is recommended that you obtain a 3x3 Rubik's cube, if you do not
already have one, e.g. from rubiks.com. This will be used to
illustrate some group-theoretic concepts in a fun, hands-on way.
(However ability to solve Rubik's cube is not a course requirement.
By the way, you can find some pretty scary stuff by searching for
"rubik" on youtube.)
Homework is due every Wednesday (except for the first two Wednesdays
and the weeks of the midterms) at 2:10 PM sharp . 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 Wednesday'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 the
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. (When I was a
student I had a deck of index cards; on each card I wrote the
statement of a theorem on one side and a summary of the proof on the
other side. Very useful!)
- 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 looking at someone else's paper.
2) All answers should be written in complete, grammatically correct
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 will
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.
There will be in-class midterms on 10/7 and 11/9, and a final exam on
- HW#1, due 9/9. Selected solutions
HW#2, due 9/16 Selected solutions
- HW#3, due 9/23 Selected solutions
- HW#4, due 9/30 Selected solutions
- No homework due the week of 10/5-10/9 because of the first
midterm on 10/7.
- HW#5, due 10/14 A couple of solutions
- HW#6, due
10/21 Selected solutions
- HW#7, due 10/28. Selected solutions
- HW#8, due 11/4. Complete solutions
No homework due the week of 11/9-11/13 because of the second midterm
- HW#9, due 11/18
Complete solutions (may contain references to what was discussed in class in 2003, sorry)
- HW#10, due 11/25.
HW#11, due 12/2: Fraleigh chapter 29, problems 4, 6, 8, 10, 12, 14
(you can take it for granted that pi is transcendental over Q), 23,
26, 30, 36, 37. Extra credit: chapter 29, problems 16, 29.
- HW#12, due 12/9: Fraleigh chapter 31, problems 8, 10, 12, 24, 25,
27, 29, 30. Complete 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.
The following is the plan for what I intend to cover, and when.
Below, numbers in square brackets refer to the relevant sections of
Fraleigh. The schedule of topics is only an approximation, so some
topics might be covered at slightly different times than listed, or
skipped if time gets short.
- 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 Rubik's cube and some symmetry groups. We will mostly
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.
- 8/26,8/28, 8/31: 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].
- 9/2, 9/4: Binary operations. Isomorphism of binary structures.
Using structural properties to show that binary structures are not
isomorphic. [2,3] (We won't follow the rather discursive .)
- 9/7: NO CLASS (Labor Day holiday)
- 9/9, 9/11: Groups. Lots of examples (not all in the
book): cyclic groups, Z_n^*, dihedral groups, the Klein 4-group,
matrix groups, symmetry groups of polyhedra, permutation groups, the
Rubik's cube group. 
- 9/14: Subgroups. Lots of examples. 
- 9/16: Cyclic groups and their subgroups.  (We will skip ,
except that we may say something about the subgroup of a group
generated by a subset of the group.)
- 9/18, 9/21. All about permutations: Every permutation is a
product of disjoint cycles (uniquely up to reordering the factors).
Every permutation is a product of transpositions. Dichotomy between
even and odd permutations. The alternating group. [8,9]
- 9/23, 9/25: Cosets, Lagrange's theorem. 
- 9/28: Direct products. Fundamental theorem of finitely
generated abelian groups (statement only).  (We will skip .)
- 9/30: Homomorphisms. 
- 10/2: Conjugation. Normal subgroups. Quotient of a group by a
normal subgroup. (Quotient groups are called "factor groups" in the
- 10/5: Review for the midterm.
- 10/7: MIDTERM #1, in class. Covers the material up to and
including the lecture on 9/30.
- 10/9, 10/12: More about normal subgroups and quotient groups.
The "fundamental homomorphism theorem". Commutator subgroup,
abelianization, center. [14,15]
- 10/14: Group actions and Burnside's formula. [16,17]
- 10/16: Rings and fields. Basic definitions and lots of
- 10/19: Integral domains.  (We will skip ; we will have
done most of this earlier.)
- 10/21: Quotient field of an integral domain. 
- 10/23: Rings of polynomials. Evaluation homomorphisms. If R is
an integral domain then so is R[x]. 
- 10/26, 10/28, 10/30: Factorization of polynomials over a field:
Division theorem for polynomials. Factor theorem. Multiplicative
groups of finite fields are cyclic. Irreducible polynomials.
Irreducibility over Z implies irreducibility over Q. (The book
defers the proof from  to .) Eisenstein's crieterion. 
Greatest common divisor and Euclidean algorithm in F[x] (cf ).
Unique factorization in F[x], using the above. (We will skip
- 11/2: Ideals and quotient rings. 
- 11/4: Prime and maximal ideals. 
- 11/6: Review for the midterm.
- 11/9: MIDTERM #2, in class. Covers the material up to and
including the lecture on 11/2, with emphasis on the material not
covered on the first midterm.
- 11/11: NO CLASS (Veteran's Day holiday)
- 11/13: More about prime and maximal ideals.  (We will skip .)
- 11/16, 11/18: Euclidean domains, PID's, and UFD's. Fun with
Gaussian integers. Any prime congruent to 1 mod 4 is the sum of two
squares. (Selection of topics from [45,46,47].) (We might not have
time for all of this, but it brings together a lot of what we have
learned about rings so it is a good review.)
- 11/20, 11/23, 11/25: Field extensions, algebraic elements,
minimal polynomials, vector spaces over a general field. [29,30]
- 11/30, 12/2: Algebraic extensions, finite extension, degree of
towers of finite extensions, algebraically closed fields, "fundamental
theorem of algebra" (statement only). 
- 12/4 (LAST CLASS): Proof that you can't trisect a 60 degree angle
using the allowed moves of classical ruler-and-compass
constructions.  (In the unlikely event that we get to this point
early, we will do a bit more with fields.)
- 12/7, 12/9: Due to a last-minute change in the academic calendar,
there is no class on these days, which are now part of a
"Reading/Recitation/Review (RRR) period". I was planning to use the
last one or two classes for review, but instead I will schedule an
optional review section and hold some extra office hours during this
period, at times TBA.
- 12/17: FINAL EXAM. Covers the entire on the course, with somewhat
more emphasis on the later material.