Math 412/612 Modern Algebra
Course Objectives

There are four main objectives of this course:
1. Developing a command for abstract axiomatic methods
2. Learning about significant topics in modern algebra
3. Mastering conventions for mathematical oral and written communication
4. Understanding some of the historical contributions of modern algbraic methods to the field of mathematics.

At the end of the course you should know the main definitions and the main theorems, but more importantly, you should understand why the definitions and theorems are formulated in the way that they are, and what they mean. You should also know the proofs for the main theorems, as well as the kinds of reasoning that go into these proofs. A part of the first objective is developing the ability to understand and construct proofs. At the end of the course you should be able to analyze an unfamiliar proof, and either verify that the proof is valid, or find errors in it. You should also be able to construct your own proofs of new or unfamiliar results. In addition, you should know how to communicate your findings orally and verbally using accepted conventions and methods of the discipline of mathematics.

Course Overview

What is Modern Algebra (also referred to as Abstract Algebra) about?  To begin to understand this, you need to think about your first experiences with algebra.  At that time, you had spent years studying operations on numbers of various sorts (whole numbers, fractions, decimals).  The new idea you learned about in algebra was operating on unknown numbers, which are now familiar to you as variables.  In essence, you learned to do arithmetic operations on variables without knowing (or needing to know) what specific numbers those variables represented.  In modern algebra we carry this abstraction a step further.

By now, you have seen that there are many different number systems that are useful in different contexts: integers, rational numbers, real numbers, vectors, matrices, and so on.  We use algebraic notation and operation symbols in all of these systems that looks essentially the same.  For example, without knowing a context, an equation such as

Ax + By = 0

might have many different meanings.  Maybe A and B are matrices and x and y are vectors.  Or maybe A and B are real constants and x and y are vectors.  Or all the variables could represent real numbers, or integers, or complex numbers.

As you also have seen, different number systems (or more properly, algebraic systems) have somewhat different properties.  If you are talking about real numbers, then AB = BA is always true.  But not if you are talking about matrices.  Similarly, when dealing with equations, x4 - 5 = 0 has no solutions if you are considering integers,  two solutions if you are considering real numbers, and four solutions if you are considering complex numbers.

In Modern Algebra we study properties of number systems in the abstract -- that is, without knowing specificially what those systems are.  Just as regular algebra allowed you to perform operations in which the exact nature of the numbers is left unknown, so in modern algebra we allow the exact nature of the operations to be unknown.

If you have studied linear algebra, you have already seen this sort of idea in action.  In linear algebra, the algebraic properties necessary for posing and solving linear systems can be formulated in a set of around 10 axioms or assumptions.  Any algebraic system that satisfies these axioms is called a vector space, and there are many different sorts of vector spaces, made up of matrices, or columns of numbers, or polynomials, or functions, to name a few.  Every vector space has common properties that can be proved using the 10 axioms.  For example, in any vector spacee there is a notion of linear independence, and of basis, and dimension.  In any vector space, the concepts of basis and dimension govern the sorts of solutions that are possible for a linear system of equations.

Indeed, linear algebra is properly a subtopic of  modern algebra.  But we will be studying other sorts of algebraic systems, with different sets of axioms, besides those for vector spaces.

What is modern algebra good for?  The short answer is that this is an incredibly powerful and indispensible tool with applications throughout mathematics.  It shows up in a central way in great theorems of number theory (like Fermat's last theorem) and applications to encryption and coding.  But on some level, asking what the topic is good for is not the point.  Mathematicians explore many areas of mathematics because the subjects are intriguing.  The search for symmetry, structure, even beauty, is a major motivation.  Modern Algebra is an area of mathematics that has been studied for 150 years for all of those reasons.  This course will introduce you to the fundamental approach and methodology of this area, and may give you broader insights about the number systems you have already studied.