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 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...
If I'm not mistaken, the instructor is a paid actor and not a mathematician! Kind of impressive if you ask me.
I'm a long subscribed Socratica enthusiast, so I'm glad that other people know about them (and appreciate them) too,
The teacher is a really good speaker and the material is explained both clearly and succinctly.
This is actually my fav YouTube channel! I can't believe I'm seeing more people on Reddit who know about them. My people! It's run by a couple of Caltech grads. They write the scripts and they appear in some of the videos (see the Calculus series and Biology series). Then there are some hosts for the other series. It's way, way more educational than other channels. I hope it gets to be more well known.