Math 325: Modern Algebra
10:30 – 11:40, Ivers 218
Concordia College, Fall 2011
Dr. Anders Hendrickson
ahendric@cord.edu
http://www.cord.edu/faculty/ahendric/
Ivers 234G
299-4742
Textbook: Modern Algebra: An Introduction, by John R. Durbin (6th ed.), supplemented
with handouts
Course Website: http://www.cord.edu/faculty/ahendric/325/
Office Hours:
When you have difficulties in this course, please come see me! The happiest part of
my job is talking about modern algebra with students. My official office hours are
Tuesday 1:00-2:00, Thursday 8:15-9:15, and Friday 1:00-2:00. Please note that these
are a minimum, not a maximum: you can also often catch me in my office at other
times. If you want to be sure to find me, just email me or catch me after class, and
we’ll set up an appointment.
Catalog Description and Prerequisite:
Introduction to basic algebraic systems: groups, rings, integral domains, and fields.
Special attention is given to the ring of integers.
Prerequisite: Math 210 – Linear Algebra.
Curriculum Goals:
Success in this course will help you “responsibly engage the world” in at least three
ways:
• You will learn about algebraic structures used by scientists to understand the
world and by engineers to influence the world through technology.
• Through the course’s emphasis on proof-writing you will learn to reason clearly
and rigorously and to communicate your thoughts with precision. This ability to
communicate mathematics is especially important to future teachers.
• In your study of groups and rings and your engagement with pure reason (guided
by intuition), you will see beauty that many people never see, and you will share
in the delight of God.
Math majors going into education may also be interested in how Modern Algebra fits
into the Minnesota teacher licensing requirements; see page 5 for details.
Attendance:
You are expected to attend all classes. Material will be covered in class that is not
covered in the book! Absences due to emergencies, illness, or college activities are
excused, but please let me know in advance so we can arrange makeup work. Absence
on the date of an exam, without prior arrangements, results in a zero score.
Modern Algebra Syllabus - Fall 2011
10:30 – 11:40
Page 2 of 6
Etiquette:
You are expected to treat me and your fellow students with respect. In particular,
• Silence your cell phone ringers and remove your headphones before class starts.
• Do not read newspapers or other extraneous material in class.
• Do not send or receive text messages while in class.
Every time I see you violating these rules, I will immediately dock 5 points
from your final course grade.
Academic Integrity (Cheating is Pathetic):
All exams, quizzes, and homework assignments are subject to Concordia’s policy on
academic integrity (http://www.cord.edu/academic/catalog/integrity.html). In
particular, quizzes and exams are to be completed without any assistance from others.
This should not need saying, but cheating will not be tolerated. It is an offense against
God, against me, and against your classmates who work honestly. If I find evidence
of cheating on a test or homework assignment, you will receive zero points on that
assignment.
Calculators:
The use of calculators is permitted during exams, but you will quickly see that they
won’t really help you much in this class anyway.
Homework: (Read this!!)
Homework will be assigned and corrected almost every class period. You may—indeed,
you are encouraged to—cooperate with classmates to complete the homework, but you
must write up your own solutions to hand in. You will probably find this class’s homework very difficult, because every assignment will ask you to discover proofs. START
EARLY!!! Finding proofs requires thinking about the problem, getting stumped, going
for a walk, thinking about subgroups and generators, eating a snack, trying to visualize
the problem in a new way, taking a shower, and then suddenly the answer will hit you.
The homework assignments will be announced in class and posted on the course website. If you happen to miss class, you are responsible for looking up the homework on
the website. Homework assignments are due at the beginning of the next class period.
(After it is collected, I may answer a few questions about the problems, so make notes
on a separate sheet of paper about what puzzled you most!)
Homework exercises are to be written neatly, and multiple pages must be stapled
together before you come to class. Do not use paper from a spiral notebook unless you
can tear off the ragged edge. Put your name on the paper. Messy, unstapled, or
ragged homework will lose points; nameless homework receives no points.
Late homework will not be accepted, except when necessitated by lengthy Concordiasponsored events such as tours. To accommodate illnesses, family emergencies, etc.,
your three lowest homework grades will be dropped at the end of the semester. The
remaining total homework grade will be scaled to 150 points.
An additional 50 points will be assigned based on your progress through the semester
in following the Elements of Style. More information about this will be given in class.
Modern Algebra Syllabus - Fall 2011
10:30 – 11:40
Page 3 of 6
Quizzes and Exams:
Definition quizzes will be given almost every Friday throughout the semester; at the
end of the semester, your total quiz grade will be scaled to 50 points.
There will be four in-class exams, currently scheduled for Monday, September 26;
Friday, October 14; Friday, November 4; and Wednesday, November 23. Absence on
the day of an exam results in a zero score, unless you have made prior arrangements
with me as soon as you know of the conflict and I agree that the conflict is worthy of a
make-up exam. (For example, family vacations, Caribbean cruises, and hockey tickets
are not valid conflicts.) The final exam is comprehensive and will be held on Thursday,
December 15 from 8:30 to 10:30 a.m. Following college policy, there will be no alternate
times provided to take the final exam. Do not make any travel arrangements to leave
campus, whether for breaks or at the end of the semester, that endanger your ability
to take the exams.
Extra Credit:
I offer three principal sources of extra credit in this course.
• First, on some of the homework assignments I will designate one or more problems
as extra credit (“XC”) problems.
• Second, the department offers a colloquium series every other Tuesday at 2:45
p.m.; you can earn up to 10 extra credit points by attending one of the colloquium
talks and turning in a two-page written report about the colloquium. (If you have
another class at 2:40 on Tuesday, you may watch the colloquium on video instead;
look for the link on the course website.) Your report should have two sections: in
the first, you should summarize the material presented; in the second, you should
give some of your own reactions to the material. The report is due in class the
Monday following the colloquium. You may earn a limit of 20 extra credit points
this way.
• Third, I will make a few extra credit projects available. These projects require
you to explore in depth subjects we may have touched on only briefly in class.
Extra credit projects will be posted on the course website. The amount of extra
credit awarded will depend both on the difficulty of the project and on how well
you complete it. You may earn up to 30 extra credit points this way. All projects
must be turned in to me by December 7.
Modern Algebra Syllabus - Fall 2011
10:30 – 11:40
Page 4 of 6
T.C. Wollan Lecture:
In honor of the centennial of Concordia’s Mathematics Department, the famous physicist and origamist Robert Lang will be coming to deliver two public lectures on the
mathematics of origami. The first talk, on Wednesday, October 19 at 7:00 p.m. in
Jones Conference Center (in Knutson Campus Center) will be for a general audience;
the second talk, on Thursday, October 20 at 4:00 p.m. in room 212 of Jones Science
Building, will be more mathematical. You should go to both!
Schedule:
Class meets every Monday, Wednesday, and Friday from 10:30 to 11:40 in Ivers 218,
with just a few exceptions. Please make note of the following special dates:
Friday, September 2
Monday, September 5
Wednesday, September 14
Monday, September 26
Friday, October 14
Wednesday, October 19
Thursday, October 20
Monday, October 24
Friday, November 4
Wednesday, November 23
Friday, November 25
Monday, December 12
Thursday, December 15
First day of class
Labor Day – but we still have class
No class – Symposium
Exam 1
Exam 2
T.C. Wollan Lecture (general), 7:00 p.m.
T.C. Wollan Lecture (math), 4:00 p.m.
No class – Fall break
Exam 3
Exam 4
No class – Thanksgiving break
Last day of class
Final exam, 8:30 to 10:30 a.m.
Grading Scale:
Grades will be computed according to the scale below. If at any point during the
semester you wish to know your current grade, please come to my office.
Homework
150 points
Elements of Style 50 points
Quizzes
50 points
Group Projects
100 points
Exam 1
100 points
Exam 2
100 points
Exam 3
100 points
Exam 4
100 points
Final Exam
150 points
Total
900 points
837–900
810–836
783–809
747–782
720–746
693–719
657–692
630–656
603–629
567–602
540–566
0–539
(93-100%)
(90-92%)
(87-89%)
(83-86%)
(80-82%)
(77-79%)
(73-76%)
(70-72%)
(67-69%)
(63-66%)
(60-62%)
(0-59%)
A
A−
B+
B
B−
C+
C
C−
D+
D
D−
F
Modern Algebra Syllabus - Fall 2011
10:30 – 11:40
Page 5 of 6
Minnesota Teacher Licensing Requirements
Objective
Subpart 3A. A teacher of mathematics understands patterns, relations, functions, algebra, and
basic concepts underlying calculus from both concrete and abstract perspectives and is able to apply this understanding to represent and solve real
world problems. The teacher of mathematics must
demonstrate knowledge of the following mathematical concepts and procedures and the connections among them:
(4) understand patterns present in number systems and apply these patterns to further investigations;
(8) apply properties of group and field structures
to mathematical investigations.
Subpart 3C. A teacher of mathematics understands that number sense is the underlying structure that ties mathematics into a coherent field of
study, rather than an isolated set of rules, facts,
and formulae. The teacher of mathematics must
demonstrate knowledge of the following mathematical concepts and procedures and the connections among them:
(2) an understanding of number systems, their
properties and relations including whole numbers,
integers, rational numbers, real numbers, and
complex numbers;
(6) geometric and polar representation of complex
numbers and the interpretation of complex solutions to equations;
(9) number theory divisibility, properties of prime
and composite numbers, and the Euclidean algorithm.
Subpart 3G. A teacher of mathematics is able
to reason mathematically, solve problems mathematically, and communicate in mathematics effectively at different levels of formality and knows the
connections among mathematical concepts and
procedures as well as their application to the real
world. The teacher of mathematics must be able
to:
(2) reason in mathematics by:
(a) examining patterns, abstracting and generalizing based on the examination, and making
convincing mathematical arguments;
(b) framing mathematical questions and conjectures, formulating counter-examples, and constructing and evaluating arguments; and
(3) communicate in mathematics by:
(b) using the power of mathematical language,
notation, and symbolism; and
When covered
§§10, 11 (Zn ); §16 (cosets, as a generalization of
the above)
§§5-8, 14-19 (Group Theory); §26 (Fields)
§5 (Examples of Groups); §9 (Equivalence Relations); §§24-27 (Ring Theory); §28 (Ordered Integral Domains); §29 (Integers); §32 (Complex numbers)
§33 (Complex Roots of Unity)
§§10-13 (Number Theory); §10 (Division Algorithm); §13 (Fundamental Theorem of Arithmetic,
Euler’s ϕ-function); §12 (Euclidean Algorithm)
Throughout the entire course
Throughout the entire course
Throughout the entire course
Modern Algebra Syllabus - Fall 2011
10:30 – 11:40
Objective
(c) translating mathematical ideas into mathematical language, notations, and symbols; and
Subpart 3H. A teacher of mathematics must:
(3) understand the overall framework of mathematics including the:
(c) examination of the same object from different perspectives; and
Page 6 of 6
When covered
Throughout the entire course
§§18, 19, 21-23 (Group Isomorphisms and Homomorphisms, Fundamental Homomorphism Theorem); §27 (Ring Isomorphisms)