I watched several videos but I still can’t understand the depths of the subject. I know its about groups and rings and other structures. While I do understand it is pure mathematics, it must have some application in other parts of pure mathematics as well, right? How is abstract algebra used in other (math) topics? I don’t understand the true depth of abstract algebra.
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A good way to think about it is taking concepts we know from math and trying to understand what they mean “in the abstract”.
For example, let’s take the concept of adding integers (positive and negative whole numbers like 0, 2, 7, -25, but not decimals or fractions).
You’ve got all of the integers, and you’ve got this concept of addition. If you add two integers you get a third integer. If you add zero, you get the same integer back. And every number has its inverse – like 7 and -7…when you add them together you get zero.
What if we took those concepts out of the integers?
We’ve got a bunch of “things”. You can perform some operation on two of those things to get a third thing. There’s one “thing” called the identity – if you apply the operation to it and another thing you always get the SAME thing back. And every thing has an inverse.
Is it possible to have those same properties hold for something other than integers?
The answer is yes, there are lots of sets of “things” and operations that satisfy those criteria.
Mathematicians call that a “group”. No particular reason, it’s just the name given to it. And there are “groups” of things that aren’t very much like numbers at all, but still have those properties.
Similar things happen if you try to study addition, subtraction, multiplication, and division and abstract those – mathematicians call that a “field”. It turns out you can define fields over sets of things that look nothing like numbers.
Studying groups and fields (and others, like “rings”) has led to all sorts of interesting discoveries. Some of those discoveries end up applying to numbers – but some don’t. We’ve discovered that many objects in the universe behave like groups, rings, or fields, so much of abstract algebra has turned out to be useful in surprising ways. But other parts of abstract algebra have turned out to be just theoretically interesting….so far.
But that’s the idea. Take concepts we know in math, and “abstract” them, to study their properties in the abstract without necessarily making assumptions about numbers.
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