Text: “Elements
of Modern Algebra”, Latest Edition, By Gilbert & Gilbert, PWS-Kent
Publisher. There will be reading
assignments from other
standard references, as well as from selected journal articles.
Course
Description:
An
introduction to modern algebra which deals
with selected algebraic structures (groups, rings, fields, etc.).
The
course stresses the axiomatic approach and the logic and method of
proof.
The
purpose of this course is to develop the
elementary properties of some fundamental algebraic structures of
groups,
rings, and fields. The theoretical
foundations
of the subject shall be emphasized, without losing sight of the rich
collection
of concrete examples and applications, which serve to motivate the
abstraction
and illuminate the theory.
Students will be
expected to
read, understand, and construct proofs.
Many of the problems or exercises are of the form, "Prove
that...." These exercises are an
important part of the course, and it is inconceivable that a student
could do
well without having seriously attempted a significant number of them.
General Class
Objectives:
1.
To
instill an
appreciation of the rich algebraic structure of our familiar number
systems.
2.
To
develop each
student's understanding of the broad scope of algebraic ideas.
3.
To
improve the
quality of communication in mathematics.
4.
To
advance each
student's ability to handle abstract mathematical ideas and appreciate
their
importance.
5.
To
provide
opportunities for the development of talents for creative thinking,
problem
solving, and writing proofs.
Grades: Grades will be determined by tests and the
final exam
as follows:
Test dates will be
announced
at least one week in advance.
|
ITEM |
POINTS |
TOTALS |
|
Chapter Tests |
100 pts |
400 - 500 pts. |
|
Assignments |
30 pts |
150 - 210 pts. |
|
Final Exam |
200 pts. |
200 pts. |
|
Total available |
|
650 - 910 pts. |
Student
Objectives:
The successful student will, by the end of the course:
1.
know
the mathematical notation, terminology,
the vocabulary, symbolism, and
basic definitions used in Abstract Algebra,
including
binary operations, relations, group, subgroup, isomorphism,
automorphism,
homomorphism, and, ring,
2.
know the statements of some of the important basic theorems for rings, groups,
and all others presented in class,
3.
be
able to
develop the fundamental properties of abstract algebraic structures,
their
substructures, their quotient structure, and their mappings
4.
be
able to use these definitions to prove simple statements
in Abstract Algebra especially those presented in class,
such as
Lagrange’s theorem, Cayley’s theorem, and the fundamental theorems for
groups and
rings
5.
be
able to prove
theorems about the structure, size, and nature of groups, subgroups,
quotient
groups, rings, subrings, ideals, quotient rings, and the associated
mappings. Solve problems about the size and composition of
subgroups and
quotient groups; the orders of elements; isomorphic groups and rings.
Topics:
Course
will include selected sections from chapters I – V.
TENTATIVE SCHEDULE OF TOPICS
|
Week |
Topics |
Sections |
|
Week
1 |
Introduction |
|
|
Week
1 |
Fundamentals
- Sets, Mappings, & Mapping Composition |
1.1,
1.2, & 1.3 |
|
Week
2 |
Binary
operations, Matrices, & Relations |
1.4,
1.5, & 1.6 |
|
Week
3 |
Test
1 |
|
|
Week
4 |
Integers
and equivalence relations; modular arithmetic |
2.1
& 2.2 |
|
Week
5 |
Integers
and equivalence relations; modular arithmetic |
2.3
& 2.4 |
|
Week
6 |
Integers
and equivalence relations; modular arithmetic |
2.5
& 2.6 |
|
Week
7 |
Test
2 |
|
|
Week
8 |
Introduction
to Groups (including a collection of examples), symmetries, dihedral
groups, and matrix groups |
3.1
& 3.2 |
|
Week
9 |
Introduction
to Groups: Cyclic Groups - classification of all subgroups of a cyclic
group, - order of an element, centralizer of an element, Isomorphisms |
3.3
& 3.4 |
|
Week
10 |
Introduction
to Groups: Permutation Groups - The symmetric group, and alternating
group, Cayley's Theorem |
3.5
& 3.6 |
|
Week
11 |
Test
3 |
|
|
Week
11 |
Normal
Subgroups, Cosets and Lagrange's Theorem |
4.1 |
|
Week
12 |
Homomorphisms
of Groups; Kernel of a homomorphism |
4.2 |
|
Week
13 |
Rings,
with examples, including polynomial rings, and matrix rings |
5.1 |
|
Week
14 |
Integral
Domains and Fields - Characteristic of an integral domain, the integers
mod p and the rationals |
5.2 |
|
Week
15 |
Test
4 |
|
|
|
Final |
|