Abstract Algebra or Topology: Which is the Better Choice for a Math Major?

[jyoungs]

Hi there,

Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?

 

[STEMucator]

How much experience do you have with calculus/analysis? How much experience do you have with algebra be it linear or abstract?

I would personally take a course in abstract algebra before I would attempt topology ( You also need a solid calc/analysis background ). The reason being is the difficulty of abstract algebra will allow you to comfortably lean into topology if your calc/analysis skills are up to par.

 

[jyoungs]

Thanks for responding and the follow up questions.

I've had the requisite calculus (two quarters of MV/vector calculus), a quarter of complex analysis (we got up to the residue theorem), and I've taken a quarter of "mathematical proofs" [introduced to logical operators, proofs and their forms (mostly elementary number theory), functions, relations, etc.]. I'd be taking this upper-div math course alongside linear algebra.

 

[STEMucator]

Thanks for responding and the follow up questions.

I've had the requisite calculus (two quarters of MV/vector calculus), a quarter of complex analysis (we got up to the residue theorem), and I've taken a quarter of "mathematical proofs" [introduced to proofs (mostly elementary number theory), functions, relations, etc.]. I'd be taking this upper-div math course alongside linear algebra.

Your analysis and proof skills sound good. I wouldn't take linear algebra at the same time as I would take abstract algebra though. If anything, abstract algebra is a followup of linear that delves much deeper so to say. This of course is up to you.

Based on what you've said though, you sound as if you would be reasonably comfortable in a topology class. Save the abstract for after you run through linear.

That's just my two cents though.

 

[Number Nine]

Your analysis and proof skills sound good. I wouldn't take linear algebra at the same time as I would take abstract algebra though. If anything, abstract algebra is a followup of linear that delves much deeper so to say. This of course is up to you.

Based on what you've said though, you sound as if you would be reasonably comfortable in a topology class. Save the abstract for after you run through linear.

That's just my two cents though.

There is absolutely no reason to require linear algebra before taking a course in abstract algebra unless the latter is, for some reason, focusing heavily on vector spaces. The material will likely be very different, and I honestly can't imagine needing any material from linear algebra except possibly the ability to multiply matrices, which is trivial to pick up with a few minutes practice.

Topology is nice, but you've never studied mathematics if you haven't studied abstract algebra.

 

[STEMucator]

There is absolutely no reason to require linear algebra before taking a course in abstract algebra unless the latter is, for some reason, focusing heavily on vector spaces. The material will likely be very different, and I honestly can't imagine needing any material from linear algebra except possibly the ability to multiply matrices, which is trivial to pick up with a few minutes practice.

Topology is nice, but you've never studied mathematics if you haven't studied abstract algebra.

I'm making my claims assuming the standard vector space curriculum (Which I'm sure happens the majority of the time without much research).

Being exposed to vector spaces and their properties will make abstract algebra a much easier experience (Considering groups, rings, fields, etc rely on these properties). Subspaces are also a very important concept to understand and will make understanding subgroups, subrings, etc much easier as well. Lots of other topics from linear algebra creep their way in too.

Also (I'll admit this is completely opinionated before I type it), learning how vector spaces work before delving into abstract algebra will give a much deeper appreciation for how things are structured.

If I'm being a bit too ambiguous, I'm referring to this:

Commutative rings $⊃$ Integral domains $⊃$ Integrally closed domains $⊃$ Unique factorization domains $⊃$ Principal ideal domains $⊃$ Euclidean domains $⊃$ Fields

 

[nonequilibrium]

Remember that the OP would be following linear algebra simultaneously with abstract algebra, so I think the argument about which one should follow which is perhaps not very important in thise case?

Concerning the choice between abstract algebra and topology: that's a tough one. Both are "important" math courses. I do think it's a bit weird to take topology without having taken analysis, but the course might be oriented towards this? @OP: does the topology course advice having had real analysis? If so, I think that's enough reason to go for abstract algebra. If not, then it's hard to choose and the choice might depend on what you plan on taking afterwards. Would you plan to take math courses again? And/or what do you expect to get from these math courses? If you would not take further math courses and you are simply looking for the course which would benefit you most in your physics career, then I would probably go for topology.

 

[jk]

Topology without analysis would be hard since analysis provides much of the motivation for introductory point set topology

 

[ModestyKing]

I'm making my claims assuming the standard vector space curriculum (Which I'm sure happens the majority of the time without much research).

Being exposed to vector spaces and their properties will make abstract algebra a much easier experience (Considering groups, rings, fields, etc rely on these properties). Subspaces are also a very important concept to understand and will make understanding subgroups, subrings, etc much easier as well. Lots of other topics from linear algebra creep their way in too.

Also (I'll admit this is completely opinionated before I type it), learning how vector spaces work before delving into abstract algebra will give a much deeper appreciation for how things are structured.

If I'm being a bit too ambiguous, I'm referring to this:

Commutative rings $⊃$ Integral domains $⊃$ Integrally closed domains $⊃$ Unique factorization domains $⊃$ Principal ideal domains $⊃$ Euclidean domains $⊃$ Fields

It sounds like if you just need to understand vector spaces, subspaces, etc., then... do the linear and abstract algebra route. Save topology for later. If you get stuck in abstract algebra because of a lack of knowledge on vector subspaces and whatnot, Khan Academy has, in my opinion, a really good set of videos for linear algebra. The first of the three sets is all about vectors and spaces, then the other sections are for matrices and all that. It'd take a few all-nighters to get all the videos done, but Sal explains it all really well. Here's the link: https://www.khanacademy.org/math/linear-algebra

Just go in order, and I stress to watch -all- the videos, even if part of it is going over something you already know.
Good luck! (And if you want to prepare for linear algebra before you take it, well, seriously, watch those videos over the course of a few weeks, take notes and/or practice, and you'll have it down. I haven't taken an official Calc 1 course yet, but linear algebra's cool! I haven't seen any Calculus yet in there).

 

[Dembadon]

... Which would benefit me more, as a student looking towards grad school?

...

Are you a mathematics major? The above statement seems to imply that these courses are electives in your program!

 

[Number Nine]

Are you a mathematics major? The above statement seems to imply that these courses are electives in your program!

Topology was an elective in my math program, which disturbed everyone (faculty included).