Teaching myself abstract algebra

Join this channel to get access to perks:

►Follow me

►My Setup:

[zachstar]

Next up, non-linear dynamics (while I finish the algebraic topology book).

[holdendinerman]

I can't believe how well this was timed. Ten minutes ago, I finished my final exam for Algebra covering this exact same textbook. I sat down to take a break and watch some YouTube, and this video happens to pop up in my subscription feed. How serendipitous!

[RatusMax]

When I went to college, I failed abstract algebra, made a C in intermediate analysis, made a C in differential equations and dropped complex analysis. I ran out of that as fast as possible and switched my major to computer science. Then the those maths started to click in my mind. I was using some concepts in there to run programs. Was it efficient? Probably not! otherwise they would teach this math in CS. Was it fun? YES!! Truth be told, I don't think I was suppose to start complex analysis in my junior year.

Stupid thing is that I bought all the math books I would need for the major. So I grabbed a book in 2020 and started reading it. It was so simple for me to understand the concepts. I feel stupid that all of this didn't click back in the day. I don't know why I didn't understand it. Although at the time I did take a huge loan out for the first time Then I moved to a cheap apartment close to the college that had bedbugs. It kept me up at night and made me like an OCD person looking at everything that moved on the wall so I wouldn't get bit. I would do rituals to clean my clothes and books so I would not bring them with me to the class. I realized I would have no money by my senior year and I didn't ask for help.

I should have applied for a full ride scholarship. My grades were pretty great before the decision to move to that other apartment. The problem is my parents never had the guidance to show me. I should have talked to someone at the school. I was young and thought that I could do everything alone. That my problems were my own to solve.




I am a different person now. I realize that not everybody is out to see me fail. If I just go out and ask for help there will be people willing to help. My childhood goal was to dual major in physics and mathematics. Then get a PhD in physics and sit the rest of my life in a research facility somewhere. I learned that my life path isn't set in stone it will change. For some reason, starting a plan in double major physics and mathematics, failing and switching to computer science has made me actually pretty good at data science. The stars aligned for me.

[xioxy]

YES I NEEDED THIS, I’m currently on this course and this is the exact book we are using. Going to enjoy the video.

[EpicMathTime]

I feel the need to point out that the operations in a ring are not any more specific than that of a group. The addition and multiplication of a ring are just names, they are not specific operations. The "addition" of a ring can be any operation in the same way that the binary operation of a group can be anything. The "multiplication" of the ring interacts with the addition of the ring through the distributive property. In other words, the terms addition and multiplication in a ring only specify their role in the distributive property relative to each other, not what the operations innately are.

For example, if we have a nonempty set X, and we look at its power set P(X), the set P(X) with the operation of symmetric difference as "addition" and set intersection as "multiplication" form a ring, simply because intersection distributes over symmetric difference. Those operations aren't addition and multiplication in any innate sense, only relative to each other, and an operation serving as addition in one ring can serve as multiplication in a different ring.

[ItachiUchiha-nsil]

Abstract algebra is awesome dude. Especially the connections with number theory.

[Adamreir]

These videos of self learning are really original, and gives a rare opportunity to hear about this from a students/ non expert. Well done!

[f-th]

It's not that the operations for a ring HAVE TO be in the usual sense; they can be anything, too (with some requirements on how they interact). It's just that we CALL them those names because it's something we're familiar with, and they behave largely in the same way.

[jow]

I'm a fan of your Zack Stars Himself channel. So I checked this out. I'd just like to commend you for being so relatable and funny in your other channel while being so mind blowingly academically gifted. It's rare to see someone with such diverse intelligence.

[PhilippeCarphin]

Oh man you have to do Galois theory next!

[mujtabasirius]

This semester I had real analysis so I thought how about I go over the course myself during winter and boom I got a notification that you made a video on real analysis. This summer I thought of self studying abstract Algebra, and then I got a notification of this video.
Considering I have topology next semester I wouldn't be surprised to get a video on that pretty soon

[luffis]

at 2:17

15x + 57y = 1 has no integer solutions and it can be shown with basic algebra (no modulo operations needed):

15x + 57y = 1
3*5x + 3*19y = 1
3*(5x + 19y) = 1
5x + 19y = 1/3

Since if x and y are integers, then 5x+19y must be an integer. However we see above that 5x + 19y = 1/3, a rational number but not an integer, giving us a contradiction.

[vladudrea]

I don't know what's different in this video, you seem... more understanding of the topics you're discussing and more in depth which, in turn, feeds the passion with which you're explaining as well. Can't wait for more of this content!

[jorada]

Abstract Algebra was my favorite math class by far. It was difficult for me, way harder than real analysis but the intuition you get from it was amazing. Like you are doing math from the ground up. Axioms and proofs to describe all the numbers. Nothing is off limits. Everything in Abstract Algebra is built off of proofs, definitions and axioms. It was pretty cool

[Mohammad-twcq]

My abstract algebra class was my favorite undergrad math class. Love how it has applications in physics as well.

[theproofessayist]

This book is a beautiful gem. Love the motivational quotes in the book too. Cheers for the JR Tolkien reference preface.

[nathanielkingsbury]

I would say, depending on the textbook you're using, linear algebra may be a more important prerequisite than you've suggested here. Michael Artin's textbook, for instance, emphasizes matrix groups (such as the group of invertible nXn matrices with real entries) as a key class of examples, especially of noncommutative groups (and similar sets of matrices as examples of noncommutative rings). Matrices also provide a particularly nice example of group action. I agree that it's not necessarily a true prereq, but I'd advise that people should be cautious depending on the choice of textbook/the professor.

[johnchessant]

It's an awesome subject! The unsolvability of a general quintic is my favorite part.

[benjamingross]

This was by far my favorite class in undergrad. I used the same book. Just candy for the brain. I think some ideas resonate with some people and others with other people. For whatever reason, I just really loved this class. It was two semesters long and covered almost the entire book. I wish I had time to do what you just did again...

[psychohosi]

I just want to add to your separation of Groups <-> Rings/Fields (which I generally agree with):

I am working in the field of calculating so called Feynman integrals. They are integrals that appear in Quantum Field Theoretic processes and are basically impossible to evaluate numerically (they love to diverge) or analytically (only possible in easy cases and too time consuming). Their calculation is crucial for more accurate calculations of elementary particle processes.

However, they can be calculated using algebraic methods that are based on the Weyl-Algebra (basically polynomials in n variables and there respective derivatives) structure of the family of all those integrals. The relevant thing here is it's ring structure and both it's ideals and it's quotient rings. In the specific approach I am working on certain Fields of rational function also appear as localizations of specific rings.

What I find interesting is that the Weyl-Algebra is more or less omnipresent at the Quantum level as it describes the relationship of space and momentum / time and energy; I think the reason that we see few applications of rings in physics is not that there are none. It's that we haven't explored it well enough yet.