Teaching myself abstract algebra 

Look I'm here

Oh my god I can't wait to see your algebraic topology video. I did a graduate algebraic topology course. I consider myself a good teacher and I would still consider teaching algebraic topology to math students to be one of the difficult subjects. I can't imagine a video that will give a general audience an idea of what algebraic topology but I can imagine that you would be up to the task of making one. I can't wait to see how you'll do it because I really have no clue what it would even take. Been a long time since I algebraic topology so I may remember it as being harder than it is.

I was thinking of algebraic geometry the whole time I wrote this comment. Then I heard you describing algebraic topology in the video and it clicked. I'm sure I'll be impressed with your explanation of algebraic topology but I won't be as impressed as if it had been algebraic geometry (I still think you can do it).

Wait... how are you able to post a comment 4 days before this was uploaded?

My only issue with Hatcher is his covering maps section. Why would you ever have a nonsurjective covering?! Other than that it's phenomenal lol

RU-vid spies on you, just as Facebook does.

And I was just going to start this book ...I m just surprised ⊙.☉

Google can read our minds now

Yo googled this texbook before. That's why youtube feed showed this video

@@victorcossio “subscription feed”, it showed up because he was subscribed, not due to him searching the book

what book did u grab?

Oof that sounds stressful, I'm glad it worked out.

I wish you success

Awesome 👍

Im kind of in the same position right now, with the whole switching from math/physics to cs thing

Very fortunate happy for you.

Enjoy.

Gallian is one of the better books out there.

Yeah. I’m currently trying to teach myself about groups so it’s very helpful.

I was wondering how do you feel about kids in 11th and 12th grade. Do you feel like dumb kids can't do 11th maths or you feel if i would have been there now, i would teach students this way or that way

Thank you, this bugged me a lot when it was said and I was hoping someone in the comments had pointed this out (and you did it very clearly and concisely imo). We like keeping things abstract after all, and "addition" and "multiplication" are more or less just mnemonic in nature.

Thanks Itachi!👍

I love the connections with topology, I'm studying schemes now.

69

+1 - In the context of rings, no specific binary operation is innately addition or multiplication. Two operations are only called addition and multiplication with respect to each other.

Well, sometimes you gotta simplify things.

@@ivarangquist9184 It's not a simplification. It's conceptually the same as how operations on groups was described.

yes

That's where things get intense, and all the different subjects come together.

Galois theory is so beautiful

Pretty sure it’s include in the book

@jow3871 Exactly. I couldn't have said it better. This is an awesome guy.

Application of algebra to physics answers a lot of disapproved theories. Love this.

Yeah, physics uses a lot of the rotation groups, which are subgroups of GL(R^3). These can be more concisely modeled with quaternion multiplication. Kinda cool

Analysis should be the same way. A good analysis text starts with the axioms for the real numbers or a construction using the rationals (and proves this construction has those properties) and everything else is developed from there. It's not really different than abstract algebra in terms of it building from the ground up. It just starts from different axioms and of course naturally develops different definitions. If you are looking for the absolute highest quality of rigour, try set theory, category theory, or model theory. I'm studying set theory right now and you would not believe the things you can construct assuming the axiom of choice. We had to construct a set in R^2 using transfinite induction such that given any distance, there was exactly on pair of points in the set that are that distance apart. Things like this make me question math because such a set is completely inconceivable. Then you assume something as harmless as a choice function and you get well orderings of uncountable sets, Banach Tarski, and transfinite induction. You can do induction over the real numbers!!! I suppose at the end of the day, all truth in mathematics is conditional, and in order to make any statements about what is actually true you must rely on faith.

I still can't figure that one out.

Yeah absolutely no complaints with this book! Best of luck!

Thanks!

Thank u

It was supposed to be my comment😂

How?

@@0mon0zz The first derivative of displacement w.r.t time is velocity, second derivative is acceleration and third derivative is called "jerk". I kid you not 😂

@@feynstein1004 Whaaaaat? lol

Yeah I should probably emphasize more but pretty much everything I say in these videos is in reference to the specific textbook I went through, I don't really have knowledge about any others (or how it's taught in university).

As a mathematically inclined physicist, I am extremely wary of abstract textbooks that use too many matrices, as it perpetuates the unfortunate truth that most physicists don't understand the distinction between a group and a representation of it. I really enjoy the Durbin book, it almost specifically tries to stay away from familiar examples, to make sure you don't accidentally impose too much structure onto the abstract concepts.

@@Saztrah Artin's book is fantastic. It covers very little representation theory, and it's motivation for using the general linear group and it's subgroups a lot are because they are very useful for many examples. All of the theory (most at least) of linear algebra is developed within the text itself. Linear algebra (with vector spaces not just Rn) is quite useful in abstract algebra, especially when you study field extensions and Galois theory. A number of proofs about field extensions can be simplified by considering field extensions as vector spaces over their base field. Then you can use powerful tools such as the dimension of the vector space. In fact, fields and vector spaces have major connections because given some field F the way a vector space V over F is defined is equivalent to V being an Abelian Group and there being a ring homomorphism from F to the group of endomorphisms of G. So in a sense when we multiply V on the left by some element in F, we are doing so in a way that preserves the structure of V. Anyways, the motivation is there to study vector spaces if you are interested in abstract algebra. The definition of vector spaces as I wrote in the last paragraph, is a very concise way of defining vector spaces using the language developed in abstract algebra. You develop intuition for objects such as fields and groups and you can now wield that intuition for studying vector spaces, and vice versa. It's use of linear algebra doesn't really relate to representation theory whatsoever, although I will say, it is somewhat difficult to distinguish a group from its representation. A representation is a homomorphism from a group to the general linear group of a vector space (that is, all automorphisms of the vector space, or in other words, the invertible matrices in the vector space, although this is dependent on your basis). What this means (by the first isomorphism theorem for groups) is that the group factored by some normal subgroup (the kernel of the homomorphism) is isomorphic to some subgroup of the general linear group, ie. as far as group structure is concerned, they are the same. I like to think of this like the group in some way captures the symmetries of the vector space. Sorry for the ramble.

I disagree -- number theory provides motivation for large swaths of abstract algebra, and was investigated earlier historically. That being said, most students today don't get exposure to the kind of elementary number theory commonly seen in competition/olympiad circles, so abstract algebra is often the first class a math major sees some of those topics, but it's an order of presentation that's only made out of necessity.

@@andresmath I concur about the historical development of the subject, but considering the modern approach to a course in Number Theory would leave most people quite helpless without abstract algebra (for algebraic number theory) or Complex Analysis (for analytic number theory), I would not suggest a course in Number Theory to precede abstract algebra. I also concur that some elementary number theory provides motivation for many topics in abstract algebra, which is why you are typically introduced to those topics within the abstract algebra course. However, the amount of elementary number theory required does not justify an entire course in Number Theory, which would surely cover much more and again would require the concepts of abstract algebra or complex analysis. Without the tools of abstract algebra or complex analysis, many of proofs in Number Theory would be inaccessible to most beginners. EDIT: I did do some additional research and found a “course in Number Theory without abstract algebra.” After reviewing the course materials, I can say it depends on one’s intentions here. If you never intend to take abstract algebra or a more advanced number theory course, then this might be the course for you. I should note that the course is not completely without abstract algebra. It just teaches those concepts within the course. So, I suppose it depends on one’s goals. EDIT2: I should also be clear that if one is interested in Number Theory, a course in abstract algebra is merely the start of that journey that provides the tools needed for that journey, much like a course in elementary algebra is necessary for trigonometry and eventually calculus.

@@stokhosursus well yeah but also there's a difference like you said between elementary number theory and Analytic number theory Analytic is the much more advanced one which abstract algebra would help in but elementary you don't need much background besides a good foundation in arithmetic

In all fairness, given his graduate background and the fact that he's already taken tons of math courses, teaching oneself abstract algebra with the Gallian text is actually pretty easy. Gallian is one of the most accessible abstract texts out there, and he organizes it really well. I would definitely encourage anyone (with some background in math) to read it!

The /sci/ wiki has a really good list of textbooks for many different fields.

I'm doing algebraic topology by Hatcher and I'm going to give non-linear dynamics by Strogatz a try. I've been craving some material with direct applications to science and engineering (that's actually discussed thoroughly in the book). After real analysis and this book I haven't gotten much of that lately lol. The next book after this one I think would be galois theory but I don't want to do that quite yet.

@@zachstar You are goanna love Strogatz's book.

@@cogitoergosum2846 Yeah Strogatz is a professor at my college. I'm planning on taking a course by him one of these years cause he's just such a great prof.

@@jeanconnery3740 Oh. How lucky you are! Being lectured by one of the best math professors out there!

You just gotta get around the twist of if, i.e. of abstraction. Once done, everything falls into place. The saying goes: once you know some, you know them all.

Galois theory is amazing. Take a book which focusses on the insolvability of the quintic like the theory was developed. It will blow your mind

Dummit & Foote is much more dense and thorough than this book. You're done with most of what you see here (excluding the Galois theory briefly mentioned via the insolvability of the quintic, which is later) in the first 9 chapters, and the coverage of modules and then vector spaces in chapters 10 and 11 is very abstract. It's a great book, but be prepared!

@@JPK314 wicked, thanks for the advice!

Yeah I would encourage you to have a more accessible text like Gallian (the one in the video) on standby when Dummit & Foote gets too incomprehensible. I've had countless friends who hated abstract with Dummit & Foote, but loved it with Gallian.

@@JM-us3fr yeah in general i've found sticking to one book is cumbersome (i used 3 books to get through my graph theory course), so your comment is good reaffirmation of that, much appreciated

@@9erik1 yea I used many books throughout that course. Try to get as many different problems in

Yeah, I think D&F is a first year graduate text, not an undergrad text. At least in most situations.

@@EpicMathTime Makes sense. I plan on using it for my thesis because it’s a great source, but I tried teaching myself with it and it was incredibly dense. My experience with Gallian was that I continued reading it even after the class, and I ended up completing almost all the (non-sage) exercises.

@@EpicMathTime what is your opinion on artin? I found it quite digestible in my intro algebra course.

Part of it is about limits. Really. When you do limits in 1D, you are talking about deltas and epsilons, in other words about intervals. In 2D, circles, 3+D, balls. But, actually, you are talking about subsets within subsets within subsets, etc. You are taking about sets that have a certain structure to them. That is part of what topology is about.

That sounds brutal lol. Galois fields came up for one chapter at the end of this book as just an introduction, can't imagine just diving into that.