Holy Cross Mathematics and Computer Science
Mathematics 352 -- Abstract Algebra II
Syllabus Spring 2008
Professor: John Little
Office: Swords 339
Office Phone: (508) 793-2274
Email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours:
- M 9 - 10 and 1 - 2,
- T 9 - 11,
- W 1 - 3,
- R 1 - 2,
- F 9 - 11,
- and by appointment
Course Homepage: http://mathcs.holycross.edu/~little/AbstractAlgebra0708/Alg2.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 this semester) it was also known that the ``general'' polynomial equation of degree 5 or greater is not solvable in this fashion.
This semester, after completing our study of some more advanced topics
in the theory of groups, we will turn to this question of solvability
of polynomials by radicals and the subject now called Galois theory
that provides the answer to this question.
We will see that associated to a polynomial f(x)
with coefficients in a field F, we can construct:
- a field E generated by F and the roots of the polynomial
f(x), and
- a group G (called the Galois group of the polynomial, or
of the field E over F), denoted
G = Gal(E/F). In concrete form,
this group acts as a group of permutations of the roots of f(x).
The structure of the group G encodes all the structure of
the field extension E/F. For instance, we will see that there
is a one-to-one, inclusion-reversing correspondence between subgroups
of G and subfields
of E containing F (the Galois correspondence).
We will see that the solvability of the equation f(x) = 0
by radicals is equivalent to the existence of a certain collection
of subgroups of G that correspond to subfields containing
the successive radicals in the formula for the roots.
To complete this line of thought, we will show that the group
G corresponding to the ``general'' f(x)
of degree n >= 5 is the full symmetric group
Sn, and that this group does not
have any collection of subgroups of the type needed
for solvability by radicals.
Along the way, we will also pause briefly to see how these
same ideas provide methods to prove that three famous
geometric construction problems from the time of the ancient
Greeks:
- the problem of squaring the circle (that is, the
problem of constructing a square with the same area as a
given circle),
- the problem of duplicating the cube (that is,
the problem of constructing a cube with twice the volume of
a given cube), and
- the problem of trisecting a general angle
have no solutions using only the straightedge and
compass of traditional geometric constructions.
To conclude the semester, we will consider a modern application
of much of the Galois theory we have considered -- the construction
of a certain class of error-control codes for the reliable
transmission of information over ``noisy'' channels.
Objectives
The major objectives of the course are:
- To continue the study of groups and prove a structure
theorem for finite abelian groups and the Sylow theorems for
groups in general.
- To develop the theory of field extensions and Galois theory
and its applications to solvability by radicals.
- 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 material from Chapters 8, 10, 11, 15, and 16 this semester.
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: Wednesday, February 27
and Wednesday, April 16. These will be given in the evening.
- Final Examination, worth
30% of the course grade. Scheduled date:
Saturday, May 3 at 8:30 a.m.
- Weekly problem sets, worth 20% of the course
grade. Notes:
- I will post complete solutions of all assigned
problems on the course homepage 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 | 1/16,18 | Structure Theorem for finite abelian groups | 8.2
|
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| | 1/21 | No Class -- MLK Day |
|
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| 2 | 1/23,25 | Finish Structure Theorem, begin Sylow Theorems | 8.2-8.3
|
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| 3 | 1/28,30, 2/1 | Sylow Theorems and applications | 8.4-8.5
|
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| 4 | 2/4,6,8 | Finish finite groups, begin extension fields | 8.5,10.1-10.2
|
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| 5 | 2/11,13,15 | Algebraic extensions | 10.2-10.3
|
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| 6 | 2/18,20,22 | Splitting fields, separable extensions | 10.4-10.5
|
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| 7 | 2/25,27,29 | More on separability, spare days | 10.5
|
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| | 2/27 | Exam 1 | given in evening
|
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| | 3/3,5,7 | No Class -- Spring break |
|
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| 8 | 3/10,12,14 | Finite fields, Geometric constructions | 10.6, Chapter 15
|
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| 9 | 3/17,19 | Impossibility proofs, the Galois group | Chapter 15, 11.1
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| | 3/21,24 | No Class -- Easter break |
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| 10 | 3/26,28 | Fundamental Theorem of Galois theory | 11.1-11.2
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| 11 | 3/31, 4/2,4 | More on the Fundamental Theorem | 11.2
|
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| 12 | 4/7,9,11 | Solvability by radicals and solvable groups | 11.3
|
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| 13 | 4/14,16,18 | Application: Error-control coding theory | 16.1-1.2
|
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| | 4/16 | Exam 2 | given in evening
|
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| 14 | 4/21,23,25 | Application: Error-control coding theory | 16.3
|
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| 15 | 4/28 | Semester wrap-up |
|
|---|
The final examination for this class will be held on Saturday, May 3 at
8:30 a.m.
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.