Learn Abstract Algebra from START to FINISH

In this video I talk about how to learn abstract algebra from start to finish. I go over some books which you can use to help you learn abstract algebra from the very basics all the way to graduate level abstract algebra.

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[Hi_Brien]

Not abstract algebra, but I wrote my first convincing proof today in LATEX! I went to all my friends not in the class and asked them to read it regardless of their feeling towards math.

[InfiniteQuest]

When you finish Lang, you do Eisenbud - Commutative Algebra with a View Towards Algebraic Geometry. Then you are the master. I have no idea how these people wrote these books. It's so much material for one person to know well enough to write a book on! I assume to write a book, you probably know way more than what's in there.

[Entropize]

If anyone is looking for a linear path:

Linear Algebra - Strang (mostly the first half)
Linear Algebra - Hoffman and Kunze (slightly more than the first half)
Abstract Algebra - Dummit and Foote immediately; get through Groups/Rings/Fields and the basics of Modules, with Lang as a reference when you need another perspective on something. I might consider rereading the group theory stuff again at some point further down the track.
Structure Theorem for Finitely Generated Modules over a PID: online lecture notes. There are a few good ones out there.
Galois Theory - Milne's notes. Stewart and Garling also work.
Commutative Algebra - Atiyah/MacDonald.
More Advanced Commutative Algebra - Eisenbud. Read as much as you need, when you need it. I have never sat down and read straight through even a full chapter, but I have read many individual theorems/proofs from it when needed to check things off.

You may need a little more on integral closures. Many texts blackbox certain important hard IC results, and I recommend "Integral Closures of Ideals, Rings and Modules" by Swanson and Huneke if you ever run into that issue.

[agh]

4:31 "An introduction to Abstract Algebra by Roy Da Bitch"

Ah yes, a classic.

[gustafa]

This class was the hardest for me, since I still wasn't comfortable with abstraction. I came back to it (by self-study) after getting used to abstraction, and it was much, much easier.

[blankino-]

Wow, I literally just bought my first abstract algebra book today!

[albieadao]

I'm loving all these START to FINISH videos. Keep making more of it. The books you have are like goldmine. Thanks for this

[pinedelgado]

Math Sorcerer does it again!!! You make learning math and the ease of learning math very, very valuable and very rewarding!!! Thank you for all the great work you've done for us!!! :) :) :) :)

[MrMegatherium]

I use to feel sad because I felt cut off from the joy of learning Abstract Algebra, because I felt I was not going to understand. I was not smart as them. But what I have found is that I can actually understand this subject. I am seeing that is actually talking about very simple things. I smile all the time because I am tasting the beauty of this subject.

[Megaluca]

For the graduate level, Algebra: Chapter 0 by Paolo Aluffi is sensational, covers all the important topics from the beginning to advanced through the eyes of basic category theory.

A story related to this book: me and my friends wanted an original copy real bad (it's like 90$ new), so we made a request to our uni's library to get one. Since we wanted to make sure they did order it, we told a lot of people to make a request for it. They probably though it was needed for a course, and got 2 copies. No way we need 2 copies. Whoops.

[nicolasbourbaki]

I'd like to add some topics and give my outline of a good path to study algebra.

First a good understanding of linear algebra is Important. Because learning linear algebra first is a great way to familiarize your self with Ideas that a common in algebra. Vectorspaces are an algebraic structure so you already encounter questions and conceps that are quit natural to ask and discover but on the other side Vectorspaces are really well behaved so answering these questions is not that hard. I don't know of a good English Book in linear algebra but the Important Topics that a good Book should Cover in my Opinion are, 1 Vector Spaces, 2 Linear Maps, 3 Determinant, 4 Eigenvalues, 5. inner product spaces and 6. the tensor products. Additionally knowing about Decomposition and normal forms of matrices and linear maps does not hurt.

After that you can Pretty much start with abstract algebra. Their are already multiple Books referenced in the Video but the most important Topics, you typical learn about in a first abstract algebra course is, an introduction to Group Theory, some basic Ring Theory that gives the prerequisites, for Field Theory and finally an introduction to Galios theory. After that you are already quit advanced and you should have witnessed some powerful results, for example the structure theorem of finite abelean groups or the fundamental theorem of galios theory.

But their are other really cool Topics in algebra, one of them is representation theory. Its basically about solving questions about groups with the help of linear algebra. You pretty much only need linear algebra as a prerequisite, so you can dive in that topic even earlier. Good Books on that are Representations and Characters of Groups by james and liebeck and the harder one but written by one of the best mathematician in the last century, linear representations of finite groups by serre.

A other Topic in that you also can immediately dive into after Linear Algebra is group theory. Although you learn a bit group theory in a abstract algebra course or book I will reference a nice part of group theory. Its combinatorial group theory. Its basically the combination of graph and group theory. If that sounds fantastic to you (wich it is) try introduction to group theory by oleg bogopolski.

Now after we have dealt with the fun part of algebra I come to the dreary part of algebra. Its commutative algebra! The study of commutative ring. Its quit technical and their is not much motivation behind the Definitions and Theorems. Its more laying the groundwork for the cool stuff than being cool. Books I Liked on the Topic, are Bosch Commutative Algebra, Introduction to Commutative Algebra by atiyah and macdonald, undergrad commutative algebra by miles reid. Books I didn't like but because the world is bad, unjust and cruel, you still have to read them(at least some parts), if you want to learn commutative algebra: Commutative Algebra by Eisenbud. Important Topics are: Modules, the different types of rings (noetherian, artenian, regular, local etc.), integral extensions, dimension theory, Valuations and Dedekind Domains, basic homological algebra(up to derived functors)

After you survived commutative algebra, fun can enter you life again. And here come the two reasons you have suffered the last months. First it is algebraic number theory. Its where many Topics you have learned before now come together to help you to understand the most basic Object in Math: The whole numbers. The main object you study in algebraic number theory are numberfields, these are finite extensions of the rational numbers. It is just really fun, because it is the first time all this stuff you have learned becomes useful (In mathematics, who cares about the real world lol). A good Book on that Topic is Jürgen Neukirchs Algebraic Number Theory. Especially the first two Chapters. The first Chapter is about understanding these Numberfields, with some really nice geometric Ideas and it culminates to a really natural proof of the quadratic reciprocity law, in just 2 lines! The other Chapter is about generalizing the theory behind the reciprocity law with the help of valuations. Wich are in it self a generalization of the norm. After that comes Class Field Theory, wich is the beginning of some the most interesting math. Like Fermats last Theorem, or the Langlands Program.

The other reason to study Commutative algebra is Algebraic Geometry. That is the study of Zeros of Polynomials in multiple Variables. Infamous for being really abstract and difficult. Im just at the beginning at understanding it but its one of the interesting topics i have encountered by now. Books on that Topic are Algebraic Geometry by Hartshorne. And the rising sea by vakil. I Will end my text now with 2 quotes on algebraic geometry:

"I can illustrate the ... approach with the ... image of a nut to be opened. The first
analogy that came to my mind is of immersing the nut in some softening liquid, and why
not simply water? From time to time you rub so the liquid penetrates better, and otherwise
you let time pass. The shell becomes more flexible through weeks and months — when the
time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! . . .
A different image came to me a few weeks ago. The unknown thing to be known
appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea
advances insensibly in silence, nothing seems to happen, nothing moves, the water is so
far off you hardly hear it ... yet finally it surrounds the resistant substance."

- Alexander Grothendieck (Who revolutionized algebraic geometry.) on his approach on math.

" As it turned out, the field seems to have acquired the reputation of being esoteric, exclusive and very abstract with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate...''

- David Mumford

Therefore obey our new algebraic overlords!!

[crggoodman]

This is my absolute favorite course I have taken in college! I’m currently in Abstract Algebra II and I love it as well. I do strongly recommend linear before this because I come across a lot of examples with matrices. We use the 9th edition Gallian book and I think it’s a great book. For some reason this course came very easily to me but I do have several friends who hated the course and struggled through it.

[lgl_noname]

It is gratifying to experience the love you have for your books.
Vous cultivez l'amour du savoir ....
pretty rare ....

[samhuang]

Algebra by Bourbaki is also classic (BUT I guess nowadays no too much beginner will learn algebra from this book). For those proficient in the Chinese language, there is a widely read and sophisticated text by Li Wen-Wei(李文威)- 代数学方法（Method in algebra）, often referred to as "LWW". In my view, LWW can be regarded as a supplementary textbook on algebra, offering concise and somewhat unconventional proofs that diverge from the usual texts. To illustrate, in demonstrating the fundamental theorem of symmetric polynomials, he employs a Young diagram, thereby rendering the proof considerably more accessible once the underlying combinatorial principles are grasped. Furthermore, the initial three chapters introduce category theory, including the concepts of Grothedieck Universe, adjoined functor, and monoidal category. As stated in the preface of LWW, the objective of the book is to establish structural coherence between concepts. It is therefore strongly recommended that those who have completed an introductory course in algebra read LWW, in order to gain an understanding of the ways in which different concepts and techniques can be structured and linked together. You could download the PDF file from Li Wen-Wei's home page, the book was based on LWW's lectures on UCAS China and could be used as introductory text for freshman according to the preface, but personally I would not suggest this as a beginning text. (Indeed I learn algebra from Lang's "Thick Dictionary").

[Juniper-]

Having just done Linear Algebra, I'm probably going to be doing a lot of abstract algebra over the summer because it feels like the natural next step. I'll probably be beginning with Gallian's book and, once I get through that entirely, I'll probably return here to find another book to look at.

[katkaka]

I always admired a people who was so special, clever at math or physics.Someone like a scientists.I dreamed so much about my own growing.

But I had to go to work and made a money for a living at very young age.

I was angry but now I am so strong and happy with my grow.
Always I was at math very bad because I believed other people how a common I am . I believed...
It was long a way..
Now I believe in myself.

Because math for me is about a path not about results.
This is the reason why I have my books with math or others, I study and read it because I want.Because everything around is a math, every science, all universe.
And I am so thankfull for a pages like this is.
And so thankfull for person like the math sorcerer is. 🙂

[GiacomoAakbr]

I bough Gallian and Fraleigh's books and they're a very good combo so far! thanks for the recommendation for sure.

[zoedesvl]

Legend says some well-known professors found that some "obvious" proof in Lang's algebra wasn't "obvious" at all, and they even wrote articles about that. Of course by "obvious proof" I mean sometimes he just didn't finish the proof and just wrote "Obvious." as a filler. Just have no idea what's going on when he was writing 🤯
Also it's just ok to find that you can't understand Lang's "examples". First year graduate cannot to be expected to understand many of them.

[edwardgraham]

Another FANTASTIC presentation by the Math Sorcerer ... Thank you for the considerable effort you put into these videos!

[dozzco]

Brilliant Video Man! I'm a Huge Fan Of C Pinter's Book On Abstract Algebra Honestly, It's just too damn good and the problems especially within the Group Theory section are Just Incredible!

Also It Covers Galois Theory And Aspects Of Number Theory Which is absolutely beautiful in its own right!