Holy Cross Mathematics and Computer Science
Mathematics 351 -- Abstract Algebra I
Syllabus Fall 2007
Professor: John Little
Office: Swords 339 (note change!)
Office Phone: (508) 793-2274
Email: little@mathcs.holycross.edu (prefered), or jlittle@holycross.edu
Office Hours: MW 1 - 3pm, TR 11 am - noon, F 10 - 11am, and by appointment
Course Homepage: http://mathcs.holycross.edu/~little/AbstractAlgebra0708/Alg1.html
Course Description
Algebra is the part of mathematics that arose historically from
the problem of solving equations of various forms. A typical
example here is the problem of solving quadratic
equations a x2 + b x + c = 0.
The well-known quadratic formula from high school
mathematics gives a general way to do this, of course.
Although the symbols we use here are relatively recent, the mathematics is so old that the original discoverers (or inventors?) are unknown. This kind of work is typical of algebra up to
about the start of the 19th century. At that time the ``state of the art'' was roughly
as follows:
- It was known that quadratic, cubic, and quartic polynomial equations could be solved in general by ``radical'' formulas like the quadratic formula, but
- On the other hand, (by work of Abel, Ruffini, and Galois that we will study in the spring of this course)
it was also known that
the ``general'' polynomial equation of degree 5 or greater is not solvable in this fashion.
The work of Abel, Ruffini, and Galois, as well as subsequent work of Cayley, Sylvester,
Hamilton, Boole, etc. was the start of what is now known as ``modern'' or
``abstract'' algebra. The basic idea of modern algebra is not only to study
how to manipulate expressions, but also to focus on the underlying algebraic
structures that allow you to make those manipulations. This is what made it possible
to prove statements like the unsolvability of the general polynomial of degree at least
5 by radicals.
To understand what this abstract approach means, consider the following.
In solving quadratic equations by the quadratic formula, we usually assume that
the coefficients a,b,c are real numbers. Then the ``usual rules of algebra'' for the addition and multiplication of real numbers - the commutativity and associativity of multiplication, the distributive law for multiplication over addition, the existence of multiplicative inverses for nonzero a,
the existence of square roots for all non-negative real numbers, etc. - are what allow us to derive the quadratic formula. Listing all these properties (omitting the existence of square roots) provides the definition of the algebraic structure known as a field.
But in fact much more insight is
gained if we study fields ``in the abstract'' and ask:
- what examples of fields are there?
- what properties do they all have?
- what special properties does the field of real numbers have? (The existence of square roots for all positive reals is one; the existence of an ordering relation that lets you define
a > 0 is another!)
- what can we say about the structure of the field ``generated by'' the roots of a given
polynomial, and how does that relate to properties like solvability by radicals?
- are there other common, useful algebraic structures besides fields, where some of the ``usual rules of algebra'' might not hold?
The last question, especially, led to what can only be called a revolution in thinking about what algebra is and what it does.
Objectives
The major objectives of the course are:
- To introduce the algebraic structures called rings and fields, and to
continue the study of groups begun in MATH 243 (Algebraic Structures),
- To show how these structures arise from many of the basic problems
of algebra and how they underlie many other topics in mathematics,
- To introduce you to some of the history and development of algebra as a mathematical
discipline,
- To continue the development of your problem-solving skills in mathematics,
- To continue the development of your skills in reading, developing, and
writing mathematical proofs.
Text
The text for the course is Abstract Algebra, An Introduction, 2nd ed.
by Thomas Hungerford, Brooks-Cole, ISBN 0-03-010559-5. All of the course readings
and most of the problems this semester will come from this text.
We will be studying the material
in Chapters 3 - 8 this semester and Chapters 9 - 11, 13 - 16 in the spring.
An interesting sidelight here: As you might guess from the dedications
of the book, the author is a HC alumnus, one of the many HC mathematics
majors who have gone on to graduate school and careers as mathematicians.
Fr. Raymond Swords, S.J. after whom our building was named, was a member
of the HC mathematics department before becoming President of the College. Prof.
Vincent McBrien was another long-time member of the department who retired
in the mid-1970's and died just last year.)
Course Assignments and Grading
The assignments for the course will consist of:
- Two in-class midterm exams, each worth 20% of
the course grade. Tentative dates: Monday, October 1 and Wednesday,
November 14. These will be given in the evening.
- Final Examination, worth
30% of the course grade. Scheduled date:
Wednesday, December 12 at 8:30am.
- Weekly problem sets, worth 20% of the course
grade. Notes:
- Because of the large size of this class, in order for me to
return your work with constructive comments in a timely manner, it
may become necessary to grade only selected problems on each assignment.
If that happens, I will always select a representative sample of
problems to be evaluated from that assignment.
But the selection will not be announced beforehand, and you
will be expected to do all of the problems in any case.
- I will put complete solutions of all assigned
problems on reserve in the Science Library after class on the date
the assignment is due. You may consult these and photocopy them
for your own use at any time if you wish.
- Because of the availability of these complete solutions,
because every effort will be made to return your graded problem
sets in a timely fashion, and for reasons of fairness,
no problem sets will be accepted for credit after the announced due date,
except in the case of a verified medical excuse. If you
are authorized to hand in a problem set late, I will ask you
sign a statement that you have not consulted the reserve
solutions in preparing your work.
- Group reports from discussion class meetings,
together worth 10% of the course grade.
I will be keeping your course average in numerical form throughout
the semester, and only converting to a letter for the final course
grade. The course grade will be assigned according to
the following conversion table (also see Note below):
- A -- 94 and above
- A- -- 90 - 93
- B+ -- 87 - 89
- B -- 84 - 86
- B- -- 80 - 83
- C+ -- 77 - 79
- C -- 74 - 76
- C- -- 70 - 73
- D+ -- 67 -- 69
- D -- 60 - 66
- F -- 59 and below.
Note: Depending on how the class as a whole is doing, some
downward adjustments of the above letter grade boundaries may be made.
No upward adjustments will be made, however. (This means, for
instance, that an 85 course average would never convert to a letter
grade of B- or below.)
If you ever have a question about the grading policy, or about your
standing in the course, please feel free to consult with me.
Schedule
The following is an approximate schedule. Some rearrangement,
expansion, or contraction of topics may become necessary. A more
detailed, day-by-day schedule will also be maintained on
the course homepage for you to consult as needed. I will announce
any changes in class, and on the course homepage.
| Week | Dates | Class Topics | Reading (Hungerford)
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| 1 | 8/29,31 | Course introduction, begin rings | 3.1
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| 2 | 9/3,5,7 | Rings | 3.2-3.3
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| 3 | 9/10,12,14 | The polynomial rings R[x], F[x] | 4.1-4.3
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| 4 | 9/17,19,21 | Irreducible factorization in F[x] | 4.4-4.6
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| 5 | 9/24,26,28 | Polynomial congruences, begin quotient rings | 5.1-5.3
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| 6 | 10/1,3,5 | Review for Exam 1, Ideals and quotient rings | 6.1-6.2
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| | 10/3 | Exam 1 (Chapters 3,4,5) | given in evening
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| | 10/8 | No Class -- Columbus Day Break |
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| 7 | 10/10,12 | First Isomorphism Theorem, prime and maximal ideals | 6.2-6.3
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| 8 | 10/15,17,19 | Groups, revisited | 7.1-7.3
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| 9 | 10/22,24,26 | Homomorphisms, Lagrange's Theorem, Normal subgroups | 7.4-7.6
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| 10 | 10/29,31, 11/2 | Quotient groups, homomorphisms | 7.7-7.8
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| 11 | 11/5,7,9 | The symmetric and alternating groups, direct products | 7.9, 8.1
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| 12 | 11/12,14,16 | Review for Exam 2, structure of finite abelian groups | 8.2
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| | 11/14 | Exam 2 (Chapters 6,7, section 8.1) | given in evening
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| 13 | 19 | The Sylow Theorems | 8.3
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| | 11/21,23 | No Class -- Thanksgiving Break |
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| 14 | 11/26,28,30 | Proofs of Sylow, applications to structure of finite groups | 8.4-8.5
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| 15 | 12/3 | Semester wrap-up |
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The final examination for this class will be held on Wednesday, December 12 at
8:30 am.
Departmental Statement on Academic Integrity
Why is academic integrity important?
All education is a cooperative enterprise between teachers and
students. This cooperation works well only when there is trust and
mutual respect between everyone involved.
One of our main aims as a department is to help students become
knowledgeable and sophisticated learners, able to think and work
both independently and in concert with their peers. Representing another
person's work as your own in any form (plagiarism or ``cheating''),
and providing or receiving unauthorized assistance on assignments (collusion)
are lapses of academic integrity because they subvert the learning process
and show a fundamental lack of respect for the educational enterprise.
How does this apply to our courses?
You will encounter a variety of types of assignments and examination
formats in mathematics and computer science courses. For instance,
many problem sets in mathematics classes and laboratory assignments
in computer science courses are individual assignments.
While some faculty members
may allow or even encourage discussion among
students during work on problem sets, it is the expectation that the
solutions submitted by each student will be that student's own work,
written up in that student's own words. When consultation with other
students or sources other than the textbook occurs, students should
identify their co-workers, and/or cite their sources as they would for
other writing assignments. Some courses also make use of collaborative
assignments; part of the evaluation in that case may be a rating of each
individual's contribution to the group effort.
Some advanced classes may use take-home
examinations, in which case the ground rules will usually allow no
collaboration or consultation.
In many computer science classes, programming projects are
strictly individual assignments; the ground rules
do not allow any collaboration or consultation here either.
What are the responsibilities of faculty?
It is the responsibility of faculty in the department to
lay out the guidelines to be followed for specific assignments in
their classes as clearly and fully as possible, and to
offer clarification and advice concerning those guidelines
as needed as students work on those assignments.
The Department of Mathematics and Computer Science upholds the
College's policy on academic honesty.
We advise all students taking mathematics or computer science courses
to read the statement in the current College catalog carefully and
to familiarize themselves with the procedures which may be
applied when infractions are determined to have occurred.
What are the responsibilities of students?
A student's main responsibility is to follow the guidelines laid down
by the instructor of the course. If there is some point about the
expectations for an assignment that is not clear, the student is responsible
for seeking clarification. If such clarification is not immediately available,
students should err on the side of caution and follow the strictest possible
interpretation of the guidelines they have been given.
It is also a student's responsibility to protect his/her
own work to prevent unauthorized use of exam papers, problem solutions,
computer accounts and files, scratch paper, and any other materials used in
carrying out an assignment. We expect students to have the integrity to say
``no'' to requests for assistance from other students when offering that
assistance would violate the guidelines for an assignment.
Specific Guidelines for this Course
Because of the large size of this class, examinations will be given
in scheduled, proctored sessions. No sharing of information in any
form with other students will be permitted during exams.
The other assignments will be the weekly individual problem
sets and about 5 group discussions.
On the problem sets, discussion of the questions with other students
in the class, and with me during office
hours is allowed, even encouraged. Consultation of other algebra
texts in the library for ideas leading to a problem solution
will also be allowed. Your final problem write-ups should be
prepared individually, however, and the wording and organization
of the writeup should be entirely your own work.
If you take advantage of any of the
options described above for consultation on the problems,
you will be required to state that fact in a "footnote"
accompanying the problem solution. Failure to follow this rule
will be treated as a violation of the College's Academic
Integrity policy.
For the group discussions, you will be expected
to work closely with your fellow team members.