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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

• 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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# What is Abstract Algebra? (Modern Algebra)

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林宜悉 posted on 2020/03/06
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