HN Books @HNBooksMonth

From what little I scrolled, I didn't see any category theory. Just some elementary group theory, point-set topology, real analysis, linear algebra etc Most of it standard material for undergrad math majors.

> And stuff that used to be simple algebra is now explained as something like 'Abelian group linear transformations'.

"Simple algebra" means different things in different professions? For example, Dummit&Foote's Abstract Algebra[0] would be considered "elementary algebra" by most mathematicians. For another example, Principles of Mathematical Analysis by Walter Rudin[1] is "elementary analysis".

[0] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/...

[1] https://www.amazon.com/Principles-Mathematical-Analysis-Inte...

The best resource I know of is the textbook I bought for the course I took on group theory and ring theory [1]. It’s pretty expensive and the exercises are very challenging but if you’re a self-motivated student, you can learn a TON of abstract algebra from this one book. You may want to review some linear algebra before you dive in, if you haven’t done so in a while. You can find solutions to many of the exercises online though I can’t vouch for their accuracy.

[1] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/...

For self study of abstract algebra, I recommend Dummit & Foote. I used it as a supplement to my undergrad algebra classes and found it to be very useful. This is in contrast to some books like Artin which leave a lot to the readers, while not necessarily bad, are sometimes difficult for self-study.

[0] https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/...

⬐ trentmb

> For self study of abstract algebra, I recommend Dummit & Foote. I used it as a supplement to my undergrad algebra classes and found it to be very useful.

Because you used it as a supplement.

⬐ dkarapetyan

I've learned from both in various undergrad and grad courses and I have to say I like Artin better. Dummit & Foote is a little too dry. I recommend Artin and Dummit & Foote as a supplement.

Most people like the Dummit & Foote book; it tends to be loaded with examples rather than a lot of dense symbolic arguments:

http://www.amazon.com/Abstract-Algebra-Edition-David-Dummit/...

When I learned it, though, it was from this Dover book, which is more affordable:

http://www.amazon.com/Elements-Abstract-Algebra-Dover-Mathem...

I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:

Abstract Algebra by Dummit and Foote http://www.amazon.com/Abstract-Algebra-David-S-Dummit/dp/047...

I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):

http://therobert.org/alg/1.1.pdf

Comments are appreciated. Better now than when I start the real journey. :)

⬐ brl

Do you want a hiking partner? How fast are you planning on going?

⬐ technoguyrob

That'd be great! I'm planning on spending a few weeks full-time on this (8 hours a day, or more if I can handle it). That should be a minimum of one section a day, but hopefully more, even a whole chapter on some days. It doesn't matter how fast, though, as if either falls behind we can just compare those solutions. Email me at technoguyrob[at]gmail[dot]com.

⬐ brl

At that pace I think I would definitely be holding you back. Maybe I'll try to tackle something a bit less steep, like Mount Fraleigh.

Have you already studied some abstract algebra, or would you mainly be learning it through this project?

⬐ technoguyrob

You can still read through my solutions and point out all my errors if you like, though. ;)

I've already had a couple classes, but I've never felt intimately comfortable with it. With several programming languages, I don't even need the occasional googling, nearly all the libraries (and of course basic syntax) are already in my head. When doing commutative algebra and Noetherian rings, I found myself looking back at ring theory for various properties about rings. Same goes for homological algebra: I had to keep going back to properties of modules. I understand it in more in terms of theorems and facts rather than mathematical intuition and maturity, and my goal is to change that around.

⬐ Xlp-Thlplylp

Note on the problem that a group G of even order contains an element of order 2. Partition G into classes {{x,x^{-1}|x\in G}. This is a partition since inverses are unique. For x\in G, call {x, x^{-1}} the class of x. At least one other element x besides the identity has a class of size one. Otherwise, the order of G would be 1 + 2*(# of classes of size 2) which is odd. Hence there is an x with x != 1 and x = x^{-1}; i.e., an element of order 2.

⬐ technoguyrob

Thanks, that's exactly the kind of comments I was looking for. That was exactly what I thought in my head, but it came out very distorted and ugly on paper. I'll update the solution later. Do you want credit?

⬐ Xlp-Thlplylp

Credit for this is optional. Another way to see obtain a partition of G is to observe that the relation x ~ y if and only if x = y or x = y^{-1} is an equivalence relation on G. The equivalence classes induce the partition above. The point is that any two equivalence classes {x, x^{-1}},{y,y^{-1}} are either disjoint or equal, and their union is G. But this is obvious (the word 'obvious' means "I thought of it.")

⬐ jfarmer

D&F was my sophomore-year college algebra textbook so all I can say is, good luck!

It gets very hard very quick and some of the questions are intentionally above virtually everyone's head.

⬐ technoguyrob

Cool, you're from the University of Chicago. What (math) field was your thesis in?

⬐ jfarmer

Chicago math students don't have to write theses. I guess they figure it's enough just to get through the curriculum.