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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Imagine you’ve done a lot of work, and proved a lot of theorems about **integers** number, and now someone says “well, integers are too restricted, can I use the same theorem on real numbers?”. Then, you have to go back through all your theorem to prove that they work on **real** numbers.
Latter, someone else says “I love your theorems, but I actually use complex numbers in my work, are your theorems still true with complex numbers?”. So you’re back at your black board and for the third time, you’re going through all your theorems for **complex** numbers.
Then, at a conference, you meet with a theoretical computer scientist, and after some talking, they say to you: “Those theorems would greatly help me, but there is just a problem, we’re not dealing with actual numbers, we’re dealing with those complex data structures. But in the idea, they work just like number, you can still add them, multiply them even if that’s a little weird to do. Do you know if your theorem would also work on those things that are almost numbers?”. So for a fourth time, you’re going through all your theorems and checking that they work for those **almost-numbers**.
And that’s a lot of useless work, because you’ve done **the exact same proof 4 times**, and peoples will keep coming up with slight variations like that are more general, or sometimes less general “What if negative numbers are not possible? Can we still use your theorem if negative numbers are forbidden?”
In my experience, the main point of Abstract Algebra is to try to make the proof once and for all. Instead of saying “I prove my theorem for integers”, you’re saying “I prove my theorem for everything that has an addition and a multiplication, and for that I need those very simple properties like n+0 = n and n*0 = 0, but outside of those simple properties you can do whatever you want. I don’t care if those are numbers, colours, data-structures, or even the weird quantum superposition of numbers that physicists keep talking about. It always works.”.
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