Introduction to Abstract AlgebraInstructor:
4:10-6:00 PM MTWTh
103 GPB
Dr. Mira Peterka
Office: 868 Evans Hall
Office hours: TuTh 3:10-4PM (in 868 Evans), W 6-7 (in 103 GPB). These hours may change.
E-mail address: peterka AT SYMBOL math.berkeley.edu
It would be helpful to have previous experience with proofs and
logic. (Appendices A-E of the textbook review some of these
topics, including the principal of mathematical induction, which
we will use early on).
Lecture Format:
According to the official class schedule, the course consists of
lecture hours and discussion hours (This distinction is made by
scheduling/the registrar purely for technical reasons involving
classroom scheduling). I will, however, be flexible about our use
of these times. Thus each days meeting will consist of a mix of
lecture, discussion, and problem solving, in no predetermined
fixed ratio.
Course Topics:
The primary objects of study will be groups, commutative rings,
and fields.
The total grade is distributed as follows:
| Homework |
%15 |
| Midterm 1 |
%25 |
| Midterm 2 |
%25 |
| Final exam | %35 |
There will be homework due on most Thursdays. The assignments
will be listed on this webpage. You are encouraged to work on the
problems with classmates, but you should write up the solutions on
your own. Do not simply copy another student's work. Typically two
problems will be carefully graded.
HW 1: Due 6/25/15
1.3: #1
1.4: #2
1.5: #1, 2, 5, 7
1.6: #1, 2, 3, 8, 14
Read sections 1.1-1.7, 1.10 of the textbook.
HW 2: Due 7/2/15
1.7: #7, 14
1.10: #2, 4, 5, 9
2.1: #2, 12, 13
2.2: #10, 15, 16
2.3: #6
2.4: #4, 6, 7, 9, 17
Read sections 2.1-2.5 of the textbook.
* Note: problems originally assigned from 2.5 have been postponed
til HW 3.
HW 3: Due 7/9/15
2.5: #4, 6, 7, 8, 9, 12, 14
2.6: #1, 2, 3, 6
2.7: #3, 4, 6, 8, 11, 12
Read sections 2.6-3.1 of the textbook.
HW 4: Due 7/16/15
3.1: #2, 3, 9, 10, 12, 13, 15
3.2: # 4, 5, 6
3.5: # 1
3.6: #1, 4, 7, 8, 13, 15
Read sections 3.2, 3.5-3.6, 5.1-5.4 of the textbook.
HW 5: Due 7/23/15
5.1: #1, 3, 5, 11, 16
5.2: # 2, 5
5.3: #1, 4
5.4: #1, 12
1.11: #11
6.1: #1
6.2: # 2, 4, 14
6.3: #10
Read sections 1.8, 1.11, 6.1-6.6, 6.8 of the textbook.
HW 6: Due 8/2/15 (note the nonstandard due date).
6.4 # 3, 8, 15
6.5 # 11, 14
6.6 # 1, 3, 4
6.8 # 5 (b, c)
7.2 # 2, 5, 8 (a,b)
Read section 7.2, 7.3 of the textbook.
There will be two midterm exams in class. Barring truly
extenuating circumstances, these exams will occur on Thursday
July 9 and on Thursday July 30. The final exam will
be in class on Thursday August 13. There will be
no make-up exams. Please bring a blue book to each of these exams.
Before each exam I will give you an idea of the format of the exam
and a sense of the sorts of problems that you will be asked.
Further information regarding the three examinations will be given
in class and on this webpage. The coverage of the first midterm
will be Sections 1.1-1.7 and 2.1-2.7 of Goodman, but also
the applications of Euler's function that we discussed in class.
The coverage of the second midterm will be Sections 3.1, 3.2, 3.5,
3.6, 5.1-5.4 (but only up to and including Corollary 5.4.5),
6.1-6.6, 6.8. The final exam will be comprehensive: the
coverage will be the sections covered on the two midterms along
with the following: 7.1-7.5, 9.1-9.5 (through Theorem 9.5.4).
Please note the following: Section 7.5 is basically a treatment of
the general theorems of Chapter 9 for subfields of the field C of
complex numbers (these fields are all automatically field
extensions of Q). Thus there is no reason to read through the
proofs of the theorems in 7.5, but to concentrate on the
statements of 7.5.1, 7.5.7, 7.5.8, 7.5.9, 7.5.11. Also, you will
want to read examples 7.5. 12-14. In class we have been carefully
proving the general theorems of the important sections 9.2-9.5
(through Theorem 9.5.4) but you won't be asked to recreate the
proofs of these on the final. You should mainly be reading the
statements of the results for understanding, and thinking about
how they relate to the examples of field extensions in Sections
7.3 and the examples of 7.5. On Monday 8/10 we should finish the
9.5. material that we began on Thursday 8/6. After that, for the
remainder of the course I will turn to calculating some examples,
including working out some of the Field Theory problems below. You
may again bring a standard 8 x 11 inch paper with (two-sided)
notes written in your hand. Further information about the final
exam will be provided in class.
Field Theory Problems:
Here are some field theory problems from the Sections 7.3 onward
that you may want to attempt as part of your preparation for the
final (note that several of these problems require more time to
work out completely than you will have in the final). If you are
"cramming", then the amount of these problems that you attempt
over the next week should probably be a function of how confident
you feel about the earlier material in the course (of course you
will want to eventually try to solve all the problems after the
final is over!).
7.3 # 9, 10, 11, 12
7.4 # 6, 10 (your choice of a, b, c)
7.5 # 6 (any of them)
9.1 #4
9.2 #3
9.7 # 1, 2
Further Comments:
Further information about the course will be given in class and
on this webpage. This webpage is under construction, so please
check for updates.
In the event of extenuating circumstances, changes to the above may occur.
It is important to attend class, read the text book, and do the homework. It is highly unlikely that you will learn the material well, or do well in the course without doing all of these things.
If at any time you develop concerns about the course, you should
discuss them with me immediately. The pace of summer school is
very rapid. Thus proper timing in dealing with concerns plays a
huge role in determining their outcomes.
Students with disabilities should speak with me right away so that appropriate accommodations can be made.
I hope that you enjoy the course!