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  • What is Abstract Algebra? Based on the name alone, you might think it’s similar to the

  • algebra course most people take in high school, just a little moreabstract. But if you

  • open a book on Abstract Algebra, youll 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 toabstractout 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 ifgroupswere so useful, it’s natural to ask: would this approach work elsewhere?

  • Soon, new abstract objects began to take shape: rings, fields, vector spaces, modulesThis

  • 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 algebrais sometimes calledmodern algebra.” And if youre

  • ever at a cocktail party with mathematicians, theyll simply call italgebra.”

  • 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...

What is Abstract Algebra? Based on the name alone, you might think it’s similar to the

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