Text:Elements of Modern Algebra - 5th edition- by Gilbert and Gilbert - PWS Publishing
General Approach: Abstract algebra is one of the most elegant and beautiful areas of mathematics. It has the elegant feel of high school geometry - concepts fit together simply and in a way which seems "just right". Algebra also gives us an opportunity to study many of the mathematical systems that are important to us in all areas of mathematics - integers, rationals, modular arithmetic, matrices, etc.
In algebra we look at the structural properties of systems which have either one or two operations, and ask what general axioms describe such systems and what we can prove from those axioms. The systems we will study with one operation are called groups (and that operation is written in general as multiplication, but sometimes, when the operation is commutative, as addition).
We will prove theorems that say things like "If G is a group, and x is an element of G, then the inverse of x is unique." Then, knowing that integers (with the operation of addition) and 2x2 matrices (with the operation of addition), etc. all form groups, we will have proved that the additive inverse (negative) of any integer or of any 2x2 matrix is unique.
The systems we will study with two operations (addition and multiplication) are called rings. As with groups, we will have general theorems that allow us to draw conclusions about many of the mathematical objects we deal with daily. The reals, the complexes, and 2x2 matrices are all examples of rings.
We will study not only rings in general, but also special rings called integral domains (where the product of two non-zero elements is non-zero --- such as the integers, but not 2x2 matrices) and fields (where you can divide - that is every element except zero has a multiplicative inverse).
For both groups and rings we will ask questions that are similar to those you have asked about vector spaces in Linear Algebra:
In Linear Algebra we asked how do you get "smaller" and "bigger" vector spaces and how could you describe the set of all possible vector spaces. (For example, for finite dimensional vector spaces over the reals, once you know the dimension the vector space is uniquely described.) We will ask similar questions about groups and rings. This will lead us to a brief discussion of one the major results of recent mathematics: the Classification of Finite Simple Groups.
In Linear Algebra you asked "What is the essential structure of a vector space (vector addition and scalar multiplication) and what are the functions which preserve that structure (linear transformations)?" Here we will ask "What is the essential structure of a group (ring) and what are the functions which preserve that structure?" (They will be called group homomorphisms and ring homomorphisms.)
Just as in Linear Algebra we proved that the composition of two linear transformations is a linear transformation, so in this course we will prove that the composition of groups homomorphisms is a group homomorphism, and the composition of ring homomorphism is a ring homomorphism. There will be many other theorems about group and ring homomorphisms in this course, which have analogies to theorems in Linear Algebra.
Although there are many analogies between the study of linear algebra (vector spaces) and abstract algebra (groups and rings), there are also significant differences. Linear algebra has much more in the way of calculation in it; in abstract algebra the emphasis is more on proving theorems. Also, linear algebra has many very important concrete applications. Although abstract algebra has important applications (to error-detecting and error-correcting codes in computer science, and to cryptography or the field of secret codes), the emphasis in this subject is on studying the abstract structures. Because of the emphasis on abstract structure and theorems this course will "feel" more like Discrete Mathematics than like Linear Algebra. And we will spend time learning how to prove theorems, both by working in a group in class and through homework.
Scheduling and Grading: There will be two tests and a final (all take-home). In addition, there will be homework problem sets collected each Friday. The homework and each of tests and final will be worth 25% of the course grade.
Outline:The course has three major parts:
a. Review of Material from Discrete Mathematics and Examination of the Integers
b. Groups
c. Rings
In each section there are problems for you to hand in (on Fridays) and problems for us to discuss in class. I collect and grade homework because it is only by working problems that you will master (mistress?) the material. There are times when you may be frustrated by a problem. You are welcome to bring it up in class. The problem –or a similar one – will be examined.
Section 1.1
Section 1.2
Section 1.3
Section 1.4
Section 1.5
Section 1.6
Section 2.2
Section 2.3
Section 2.8
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Section 3.5
Section 4.1
Section 4.2
Section 4.3 – read only
Section 4.4
Section 4.5
Section 4.6 – read only
Section 4.7 – read only
Section 5.1 # 1, 2e,-f, 7ace, 16, 20, 25, 37
Section 5.2 #1, 2, 8, 10 (note: this is the converse of what we proved in
class), 15ad
Section 5.3 #11
Section 5.4 = read only; you may choose to read some of the problems, but
that is optional.
Chapter 6
Please note that we have discussed all the material in Chapter 5, so you are
able to do these problems. I have discussed some of the material in Chpater
6 sections 1 and 2. It would certainly help you to read those sections.
After Monday you should be able to work on those problems:
Section 6.1 #3, 6, 11; (We will do 12 in class) 13a, c, 18, 22, 23 (might do
this one in class)
Section 6.2 #6, 7, 9, 12, 15, 18
Section 6.3 - Read only. For giggles you might want to try #10. (You might
also want to prove that every Boolean ring is commutative.)
Section 6.4 will probably be read only.
If time allows, we might talk a little about the beginning of Ch. 8.