Abstract algebra

So over the last year or so I've really started getting into simulations and numerical analysis, which I never thought I would enjoy but hey here I am. I want to understand abstract algebra better, and just like how making physics simulations has really helped me understand physics principals better I want to do some sort of coding project with abstract algebra to understand abstract algebra concepts better. Problem is, when I try looking up "Computational group theory" or "computational abstract algebra" I dont find many useful resources or places to go to help scratch this itch. Im hoping some of you might be able to help me out here by pointing me in the right direction. You know, half the time we cant seem make progress because we don't know what to search for. Im hoping someone here can help tell me what to search for.

I took a full credit (two semesters) abstract algebra in my undergraduate many years ago. The course covered groups, rings, and fields and progressed in that order. We started with groups, learned about their properties, then learned about rings which added some properties to groups, then talked about fields, which added some more properties to groups and rings. My professor at the time said that he's seen it taught this way, and in the reverse order: starting with fields, then taking away properties to end up with rings and then with groups.

Which way did you learn? And did you like this?

Everything about this class just hurts, it feels like it is like goes completely against all my strengths in math. I’m someone that just doesn’t play well with just the raw computational parts of math. I got through real analysis and every other math/physics class i’ve taken and felt it was pretty easy, most of the theorems and results felt pretty intuitive and “just made sense.” I just don’t feel the same way at all about Abstract Algebra, and it feels like I have to jump through a billion mental hoops to reach the results. i can’t just “see” it in my head like i could with analysis. On a separate but somewhat related note if anyone has a good framework that allows them to really integrate a conceptualization of the topic into their mind and make it feel intuitive PLEASE HELP ME. I ABHOR the feeling of not fully getting and having the intuition for these concepts and hate just feeling like im memorizing results to pass a test, i really want to get this.

EDIT: Thanks everyone for everything. I got a lot of new things to look at and some offers for help, much appreciated. On another note, this post was motivated by a midterm that i took this morning which traumatized me, but results are in and while my grade isn't great in an absolute sense, i did well relatively in my class! we are so back ladies and gents :D

Hi, I’ve started self-learning Math and, after going through Michael Spivak’s marvelous Calculus, and David C. Lay’s Linear Algebra, I wanted to start Abstract Algebra. I’ve done some research and I’m hesitant between Fraleigh’s A First Course in Abstract Algebra, Dummit and Foot’s or Michael Artin’s Algebra. Which one should I go first? Any other suggestions? I’m listening.

Hi all, I have been interested in Abstract Algebra. Apart from the beautiful theoretical side, what are some of it applications? Thank you in advance.

So I just stumbled upon a whole new world of music theory, with neo riemannian music theory, the tonnetz, and all of that weird math and geometry stuff. It just feels excessive, what is the point of an omni fold in music? What is a tonnetz for? it just feels like a complicated way to form chords. Now I am pretty well versed in theory, but this has me confused, it’s just so, well, abstract. Can someone give me the basics and the purpose? Disclaimer: I have just stumbled across this so sorry for my naïveté.

Does anyone know of any good online resources (books, lectures, notes) that would be good for me to self study abstract algebra? I only have math education up to calculus but I’m ok being in a little over my head. Thanks!

Math background: good at high school maths and the maths modules in my engineering degree. Know very little about pure maths.

What's the go to textbook for introducing someone to groups, rings, fields. Like the equivalent of Spivak for analysis?

I asked ChatGPT for recommended books to self-teach abstract algebra for someone who has no prior knowledge and wants to start from the basics, it gave me these: "A First Course in Abstract Algebra" by John B. Fraleigh "Algebra" by Michael Artin "Abstract Algebra: Theory and Applications" by Thomas W. Judson "Introduction to Abstract Algebra" by Keith Nicholson

For someone who wants to learn it because it's a precursor to cryptography, like me, it told me the first one, the one by John B. Fraleigh, would be the best. Can i go with it or do you know anything more appropriate?

And should i finish learning calculus before i start with abstract algebra/number theory/cryptography? Basically i know all maths up to derivatives and the very basics of integration, but very little linear algebra. Sadly they don't teach you that stuff in grade 11 yet.

I plan on going to graduate school within the next year to pursue a Ph.D. I'm interested in several areas of mathematics but I'm particularly interested in abstract algebra(like Lie theories or Clifford/Geometric algebras) and applied mathematics (like nonlinear dynamics or numerical analysis). I'm curious to see what kind of research people are currently working on in fields that intersect the two or that are adjacent. Or if there's even any research being done at all.

Mathematical physics is an obvious one with physical gauge theories being based on Lie groups/Lie algebras, applications of Clifford/geometric algebras and geometric calculus to physics, and operator theory/operator algebras applications in QFT and quantum information theory. I really like physics and am interested in mathematical physics, but I don't really have a background in it besides what I've taught myself.

I'm aware of topological data analysis and I've also heard of , a numerical integration technique that uses Lie group actions on manifolds.

I am looking for next book after Dummit and foote Abstract Algebra. Most probably people start Commutative Algebra by athiya. But I am looking for general abstract algebra book or a book which reviews undergraduate Abstract fastly, then introduce a particular branch of Algebra (like Homological Algebra, representation theory etc).

After a bit of research I found these books • Advanced modern algebra by Rotman (part 1) • Abstract Algebra by Grillet. • Algebra a graduate course by Issac Martin • Algebra 3rd edition by Lang

My confusion is which book should i pick, just after Dummit and foote ( here i mean after covering basic groups rings fields from D&F)?

Furthermore, If any recommendation of book rather than general abstract algebra, then please give that too. Like : •Group theory by Rotman •Non Commutative rings by Lam •Field and Galois theory by Mirandi •Homological Algebra by Rotman Etc

Just after Dummit and foote. In other 6after covering standard undergraduate Abstract Algebra. Thank you.

Hello you all,

I have dabbled both with Haskell and abstract algebra and I’m wondering if there’s a relationship? If there is could you please send me any links to information I can use to learn about this? Thanks

So it’s pretty much the title. I have to take Abstract Algebra to complete my math minor. I am really struggling with it and we are only week 5. Our first exam is next week and I’m terrified. I think a reason I’m struggling with it is because I haven’t heard or been shown any real world applications and it feels like the deeper we get the farther away we get from any math I’ve seen before.

So, any advice on Abstract Algebra at all?

Edit: Thanks guys! These responses have helped!

Edit 2: I just really want to say thank you to all of you. A lot of you guys have really helped and have helped me change my perspective on this class as a whole. I was feeling really discouraged but I’m starting to get the hang of it. I had a talk with my professor too that really helped. I’m feeling better about my first exam coming up.

Hello I am self studying abstract algebra. I'm finding it very boring. I'm sure it's fun though because a lot of people know about it. Is there any cool relationship or application of it that I should know of to make it more interesting? I usually just study math by skipping to the fun stuff and learning about everything else as I go.

Please suggest me some good and popular books on abstract algebra published by Springer (for self-study purpose, ideally alternatives to Dummit & Foote).

Why Springer? You may not believe it, but because books published by the other publishers are just too expensive compared to "discounted" Springer books. Last year I managed to get a bunch of Springer textbooks for just 20€ each.