There are four main objectives of
- Developing a command for abstract axiomatic methods
- Learning about significant topics in modern algebra
- Mastering conventions for mathematical oral and written
- 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
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
you should be able to analyze an unfamiliar proof, and either verify
the proof is valid, or find errors in it. You should also be able to
your own proofs of new or unfamiliar results. In addition, you should
how to communicate your findings orally and verbally using accepted
and methods of the discipline of mathematics.
What is Modern Algebra (also referred to as Abstract Algebra) about?
To begin to understand this, you need to think about your first
algebra. At that time, you had spent years studying operations on
of various sorts (whole numbers, fractions, decimals). The new
you learned about in algebra was operating on unknown numbers, which
now familiar to you as variables. In essence, you learned to do
operations on variables without knowing (or needing to know) what
numbers those variables represented. In modern algebra we carry
abstraction a step further.
By now, you have seen that there are many different number
are useful in different contexts: integers, rational numbers, real
vectors, matrices, and so on. We use algebraic notation and
symbols in all of these systems that looks essentially the same.
example, without knowing a context, an equation such as
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
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
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
mathematics. It shows up in a central way in great theorems of
theory (like Fermat's last theorem) and applications to encryption and
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.