In the most basic sense, what is abstract algebra about?
Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields."
I find this, however, to say the least, not very informative. What do they mean by abstract algebraic structures? Along these lines, what are groups, rings, and fields then?
I've been told by a friend that groups, essentially, are sets of objects, although, this still leaves me wondering what he means by objects (explicitly).
I don't need anything rigorous. Just some intuitive definitions to give me some direction.
Thanks!
Best Answer
We learn math with numbers early on. We learn how to apply operations to numbers to get new numbers. We learn rules, and consequences of those rules. All of that is pretty straightforward.
But, the real numbers are not the only things we might want to examine in detail. The properties of how elements interact under operations is a more general, abstract notion of what we do with numbers when we do algebra.
For instance, maybe we want to examine what a shape looks like if we rotate it around. Maybe you run a supply chain, and you need to build 4 widgets, but only some of those widgets need to be built in a certain order. Could you re-arrange things to make it more efficient? Maybe we want to explore structures that have a fundamental periodicity, like the time of day.
Over time, we have constructed concepts of structures that elements can belong to, and notions of operations on these structures. These structures -- groups, fields, rings, monoids, modules, vector spaces, etc. -- don't have a natural set of rules, per se. We make up those rules (aka axioms), but we have found that many natural concepts adhere to those rules.
This is all well and good but somewhat useless until you learn about isomorphism. Exploring what a group is or what a ring is is fine. But the richness of abstract algebra comes from the idea that you can use abstractions of a concept that are easy to understand to explain more complex behavior! Adding hours on a clock is like working in a cyclic group, for instance. Or manufacturing processes might be shown to be isomorphic to products of permutations of a finite group.
Abstract algebra is what happens when we want to explore consequences of rules and properties on collections of objects of any type -- hence the term "abstract!"