How was your abstract algebra course structured?

Question

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?

Answer 1

We did a lot of group work

Answer 2

My undergrad algebra was pretty much the same as yours. The first course covered groups with a focus on finite groups, the basics of ring theory including the abstract Chinese Remainder Theorem, PIDs, UFDs, and Euclidean Domains. The second course repeated the same content on ring theory using the same text, reviewed linear algebra for a week, briefly introduced modules, covered the classification theorem for finitely generated modules over a PID without proof, covered basic field theory, and did a brief survey of Galois Theory.

I can't see how treating these subjects in the reverse order would really work. Field theory requires a lot of the results about polynomial rings because extensions are constructed as quotients of polynomial rings, so I don't see how it could be taught to students who don't know the basics of ring theory. I also remember using group theory (at least Lagrange's Theorem) when studying finite fields although that can probably be avoided. Additionally if fields are taught before groups then basic Galois theory cannot be taught right after field theory is introduced, and that seems strange to me.

Answer 3

We called it Modern Algebra but got taught the same things. We also started with groups and and learned their properties. I never thought of it being taught in the reverse way. Might of gotten a bit more confusing

Answer 4

Well I'm part of the ones that learnt it in the "reverse way". For me wasn't confusing or something. My first course on abstract algebra was poorly motivated so I really failed to see what was the point of Algebra and what mathematicians that focused their research on algebra were doing. I still didn't dislike it  but I think the order wasn't a problem.

Answer 5

My course was different. We learned rings, fields, and groups in that order. I wish there had been more emphasis placed on groups and perhaps groups be taught first. I thought the structure helped build up to what a group is, but it felt more like an afterthought when really it should have been the main focus. Topics like Galois theory were omitted all together from the course material (perhaps saved for the second series in the course). In classes since, I’ve constantly had to work around group properties while field properties make a difference.

Answer 6

Sylow's theorems should not be taught and instead more attention should be placed on homological algebra.

Answer 7

My linear algebra experience from undergrad to graduate algebra went like this:  
1. starting with proof based linear algebra, vector spaces, matrices, quotient spaces, etc  
2. basic group theory, isomorphism theorems, quotient groups. Sylow theorems could be here but normally are deferred till a later course in group theory at the graduate level.  
3. rings, modules, and module decomposition theorems (including decomposition of finite abelian groups)  
4. finite fields and Galois theory.  
5. everything above but from a more advanced and categorical perspective, like using Lang as a reference.    


I'm probably missing something lol, but this feels like what we learned through 3 courses (2 undergrad and one graduate)

Answer 8

My very first algebra course used undergrad Hungerford which starts with rings and then does groups. We had a separate course that did fields along with Galois theory.

Answer 9

What you describe is how I learned it. This was from Gallian’s text. Hungerford’s undergraduate text is structured the other way around.

I tend to not think of rings building on groups, however. They are very different animals to me. But then I’m an Analyst.

Answer 10

We started at the beginning of hersteins book, topics in algebra, and went to the end