algebra course most people take in high school, just a little more… abstract. But if you

open a book on Abstract Algebra, you’ll be in for quite a shock. It looks nothing

like the algebra most people know about. So to help you understand the subject, let’s

go back in time …

The year is 1800, and for some time now, people have known how to solve linear equations,

quadratic equations, cubic equations and even quartic equations. But what about equations

of higher degree? Degrees five, six, seven and beyond?

A young teenager named Évariste Galois answered this question.

And to do so, he used a tool that he called a “group."

Around this time, Carl Friedrich Gauss was busy making discoveries of his own. He ironed

out a new technique called modular arithmetic which helped him solve many problems in number

theory. Modular arithmetic shared many similarities to the groups used by Galois.

The 1800s also saw a revolution in geometry. For more than 2,000 years, Euclid dominated

the scene with his book The Elements, but mathematicians began to realize there are

other geometries beyond the one devised by the ancient Greeks. It didn’t take long

before groups were found to be a useful tool in studying these new geometries.

It soon became clear that groups were a powerful tool that could be used in many different

ways. So it made sense to “abstract” out the common features of this tool used by Galois,

Gauss, and others into a general tool, and to then learn everything about it.

Thus, group theory was born.

And if “groups” were so useful, it’s natural to ask: would this approach work elsewhere?

Soon, new abstract objects began to take shape: rings, fields, vector spaces, modules… This

didn’t happen overnight. It took years of hard work to find the right definitions. Too

specific, and they wouldn’t be very useful. Too general, and they would be kind of boring...and

NOT very useful. But eventually the right definitions were identified. Altogether, they

form the subject we now call Abstract Algebra.

At first glance Abstract Algebra may not seem very applicable to the world around us. But

it’s a young subject, and its usefulness continues to grow. Every year, new uses of

Abstract Algebra are found, and not just in mathematics. Physics, chemistry, computer

science and other areas are discovering just how useful abstract algebra can be.

Quick note - “abstract algebra” is sometimes called “modern algebra.” And if you’re

ever at a cocktail party with mathematicians, they’ll simply call it “algebra.”

So, when are you ready to begin learning Abstract Algebra? First, you really need to know the

more familiar algebra, but the most important requirements are these: mathematical experience

and mental maturity. Have you seen many mathematical proofs before?

Are you able to think VERY abstractly?

If so, then get ready, Abstract Algebra will challenge you like never before...