Math 113: Abstract algebra
UC Berkeley, Spring 2003
Instructor
Michael
Hutchings
hutching@math.berkeley.edu
Office phone: 5106424329.
Office: 923 Evans.
Office hours: Tuesday 9:30  11:30, 1:00
 2:00 (subject to change or temporary rescheduling), or by
appointment.
Office hours the week of 5/19 to 5/23: Tuesday, Thursday, and Friday,
3:00 to 5:00 PM.
Graduate student instructor
In addition to my office hours, Eli Grigsby will hold office hours for
all Math 113 students, in room 891 Evans, on Wednesdays from 103pm and
Thursdays from 811am and 46pm.
Spacetime coordinates
The course meets in 70 Evans on Tuesdays and Thursdays at 8:00 AM.
Course goals
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 encompasses all of this and
much more, and delves deeply into questions of ``what is going on''
with algebra in general. 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 form a very
interesting 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 develop 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.
Textbook
The textbook for this course is John B. Fraleigh, A first course
in abstract algebra , 7th edition, AddisonWesley.
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, 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 abstract algebra texts out there, and if you want
some additional perspectives you might try browsing through these in
the library or your favorite bookstore.
Equipment
It is recommended that you obtain a 3x3 Rubik's cube, if you do not
already have access to one, e.g. from rubiks.com. This will be used to
illustrate some grouptheoretic concepts in a fun, handson way.
More about Rubik's cube
Rubik's cube has been extensively studied by cube enthusiasts over the
last 25 years, and I recently discovered this
webpage which has links to enormous amounts of amazing information
about it. For example, section 6 of that webpage includes links to
computer programs which in a matter of days can solve a scrambled cube
with the absolute minimum number of turns, and which within seconds
can come within a couple of turns of the minimum! As far as I know the
minimum is always at most 22 or 24 turns (depending on whether you
count a 180 degree turn of a face as one turn or two), and usually a
bit less, much shorter than human algorithms for solving the cube
which may require 60 turns or more. There is a beautiful trick using
a little group theory (namely cosets, which we will learn about) which
allows the computer to quickly find a nearly minimal solution, and
also to dramatically prune the search tree in order to find a minimal
solution in a reasonable amount of time.
Notes on how to write proofs
I put together some old notes which give a very basic
introduction to mathematical proofs. Here they are: pdf postscript.
Approximate syllabus
The following is a rough, tentative plan for the topics we will cover.
What we actually end up doing in class will be listed below after each
lecture.
 Preliminaries. We will begin with a brief review of some
preliminary topics, including sets, relations, induction, and some
very basic number theory. Some of this material is in section 0 of
the book, some is scattered throughout random later sections, and some
is not in the book at all. (Some is covered in the notes 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 do some
additional group theory from Part VII as time permits.
 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 Part IV and some of Parts 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, and maybe a bit of
Part X. We will develop enough machinery to prove that one cannot
trisect a sixty degree angle with a ruler and compass.
What we actually did in class
 1/21: About the course; review of sets and functions. (Some of
this is in Section 0 of Fraleigh.)
 1/23: Equivalence relations
and modular arithmetic (see Section 0). Review of proof by induction
(see the above notes).
 1/28: Strong induction and existence of
prime factorizations. The wellordering principle and the division
theorem (in the notes). Uniqueness of prime factorizations.

1/30: Conclusion of preliminaries: greatest common divisor, euclidean
algorithm, and solving ax = b (mod n). Introduction to binary
operations (see sections 1, 2, and 3 of Fraleigh).
 2/4:
Isomorphism of binary structures (see section 3).
 2/6: Groups
(see section 4). Introduction to nontrivial examples including Z_n^*,
the Rubik's cube group (regarded as a subgroup of S_48), and dihedral
groups.
 2/11: More examples of groups including O(n), SO(n), the
Klein 4group, and symmetry groups of polyhedra. Subgroups (see
section 5). Examples including subgroups of D_2, Z_4, and D_3, and
some of the many subgroups of the Rubik's cube group.
 2/13: The
subgroup of a group generated by a subset of the group; examples.
Cyclic groups. Every cylic group is isomorphic to Z or Z_n.
Subgroups of cyclic groups are cyclic. The subgroup of Z generated by
a and b is generated by gcd(a,b). (See section 6 and some of section
7. We're not doing Cayley graphs, but you can read about them for
fun.)
 2/18: Permutations in detail. Every permutation is a
product of disjoint cycles (uniquely up to reordering of the factors).
Every permutation is a product of transpositions. Definition of even
and odd permutations. (See section 9 and some of section 8.)

2/20: Proof that no permutation is both even and odd. The alternating
group. Statement of Lagrange's theorem and proof of basic
corollaries. (See section 9 and some of section 10.)
 2/25:
Review for the midterm. Sam Lloyd's "15" puzzle. Cosets (see section
10).
 2/27: We had the first midterm.
Here are solutions. Here is the score distribution.
 3/4: Proof of
Lagrange's theorem (section 10). Direct products (section 11).
Statement of the fundamental theorem of finitely generated abelian
groups. (A proof, which is elementary but complicated, is given in
section 38.) By the way, the book I recommended in class is
Godel, Escher, Bach by Douglas R. Hofstadter.
 3/6: Homomorphisms
(section 13). Normal subgroups. The quotient of a group by a normal
subgroup (section 14). (Quotient groups are called "factor groups" in
the book.)
 3/11: More about normal subgroups and quotient
groups. The "fundamental homomorphism theorem". Examples. (See
sections 14 and 15.)
 3/13: Commutator subgroup and
abelianization. Converse of Lagrange's theorem is false (section 15).
But Cauchy's theorem provides a partial converse. Statement of the
Sylow theorems (section 36). (We don't have time for more advanced
group theory in this course because we need to do the basics of rings
and fields. But you might enjoy reading sections 36 and 37.)

3/18: Rings. Lots of definitions and basic examples. (See section
18.)
 3/20: Integral domains. The field of quotients of an integral
domain. (See sections 19 and 21.) (You might also want to read
section 20 for fun; we did some but not all of this material before.)
 3/25 and 3/27: Nothing (spring break).
 4/1: The ring of polynomials R[x]. If R is an integral domain
then so is R[x]. A polynomial determines a function R > R, but
the function does not always determine the polynomial. Evaluation
homomorphisms R[x] > R (need R commutative). Division theorem for F[x]
where F is a field. The factor theorem. Algebraically closed fields.
(See section 22 and some of 23.)
 4/3: Irreducible polynomials. Over a field, a polynomial of
degree 2 or 3 is irreducible iff it has a root. If f in Z[x] is
irreducible over Z then it is irreducible over Q. (The book doesn't
give the direct proof that we gave, although
an elegant more general statement is proved in section 45.)
Corollary: rational root theorem. The Eisenstein criterion (see
section 23).
 4/8: If p is a prime number then Z_p^* is cyclic (see section
23). If F is a field of characteristic zero then there is a unique
nonzero homomorphism from Q to F. Review for the midterm.
 4/10: We had the second midterm. Here are the questions with solutions. Here is the score distribution.
 4/15: Greatest common divisor and Euclidean algorithm in F[x].
(The book doesn't really do this, although there is a generalization
in section 46.) Unique factorization in F[x], using the above. (For
a different proof see sections 23 and 27.) Z[sqrt{5}] is not a
unique factorization domain. (See Example 47.9 and the definitions at
the beginning of section 45.)
 4/17: Ideals and quotient rings (section 26). Started on prime
and maximal ideals (section 27).
 4/22: We covered the rest of the material in section 27, but I
tried to organize it a little more clearly than in the book. Next we
will be doing Part VI (Extension fields), which will be the climax of
the course.
 4/24: Field extensions, algebraic numbers, minimal polynomial,
etc; see section 29.
 4/29: More about simple algebraic extensions, see section 29.
Review of vector spaces over an arbitrary field, see section 30.
 5/1: Algebraic extensions, finite extensions, degree of towers of
finite extensions (see section 31).
 5/6: We proved that you can't trisect a 60 degree angle with a
ruler and compass, cf. section 32.
 5/8: We proved the classification of finite fields, see section
33. This is a long and not so easy proof, but it brings together a
lot of the concepts we have been studying lately so it is a good review.
 NOTE: on 5/13 there will be a guest lecture by Prof. Tara Holm.
Although new material in this lecture will not be on the final, the lecture
will bring together more of the previous course material, so it is
strongly recommended for review. I will be out of town for the week
of 5/12 to 5/16, but I will hold office hours the following
week, on Tuesday, Thursday, and Friday from 3:00 to 5:00 PM.
Homework assignments
 HW#1, due 1/30. pdf ps Selected solutions. pdf
ps
 HW#2, due 2/6. pdf ps
Selected solutions. pdf ps
 HW#3, due 2/13. pdf ps
Selected solutions. pdf ps
 HW#4, due 2/20. pdf ps
Solutions. pdf ps
 No homework due 2/27 because of the first midterm. The first
midterm will cover what we do through 2/20.
 HW#5, due 3/6. pdf ps
Selected solutions. pdf ps
 HW#6, due 3/13. pdf ps. Sorry, no solutions this week.
 HW#7, due 3/20. pdf ps
Selected solutions. pdf ps
 No homework due 3/27 because of spring break.
 HW#8, due 4/3. pdf ps.
Solutions. pdf ps
 No homework due 4/10 because of the second midterm. The second
midterm will cover what we did through 4/3 (except that the Sylow
theorems will not be covered).
 HW#9, due 4/17. pdf ps.
Solutions. pdf ps
 HW#10, due 4/24. Section 26, problems 3, 10, 15, 16c, 22, 30,
31. Section 27, problems 6, 14, 15, 16. Section 45, problem 10.
Extra credit: section 27, problem 38.
Solutions. pdf ps
 HW#11, due 5/1. Section 27, problem 32. Section 29, problems 4,
6, 8, 10, 12, 14, 16, 23, 26, 29, 30, 36, 37.
Solutions. pdf ps
 HW#12, due 5/8. Section 30, problems 10, 15, 26. Section 31,
problems 8, 10, 12, 19, 23, 24, 25, 30, 31, 34, 36.
Solutions. pdf ps
 Final exam review.
Homework policy
Homework will be due on Thursdays at 8:10 AM 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!) Assignments will
be posted below on the preceding Thursday night. 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. However it is OK if
you miss the deadline once or twice, because your lowest two homework
scores will be dropped.
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, and 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 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." 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, and resist the temptation to
use phrases such as "it is easy to see that...". You can of course
cite theorems that we have already proved in class or from the book.
Exams and grading
There will be inclass midterms on 2/27 and 4/10, and a final exam on
5/24.
Makeup exams will be allowed only in extreme circumstances, and may
be in a different format such as an oral exam.
Please note that exam grades cannot be changed 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, and
regrading one student's exam would not be fair to everyone else.
The course grade will be determined as follows: homework 30%, midterms
15% each, final 40%.