Questions tagged [abstract-algebra]

For questions about the study and teaching of abstract algebra, including topics such as groups, rings, fields, and vector spaces.

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12
votes
1answer
2k views

Is MacLane and Birkoff's “Algebra” suitable today as either an undergraduate or graduate text in abstract algebra?

I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's Algebra at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the ...
30
votes
4answers
3k views

Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra or Hodge et al'...
19
votes
8answers
821 views

What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty ...
12
votes
4answers
421 views

What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...
21
votes
10answers
813 views

What makes cosets hard to understand?

I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the ...
4
votes
5answers
3k views

Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
17
votes
2answers
526 views

Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler ...
9
votes
7answers
1k views

Galois Theory: necessary?

I noticed the discussion of whether the teaching of Galois Theory is necessary on MathOverlflow. Here at LSE, everything we teach in mathematics should have some application to the social side of life....
6
votes
4answers
291 views

Content for a two-quarter class on abstract algebra

I am going to teach a two-quarter sequence on abstract algebra at a mid-size american public university. Ideally, this course would introduce groups, rings, and fields, then end with some ...
1
vote
0answers
107 views

Online open-course-ware that uses Maclane's book “Algebra”

I am struggling with that book which I find to be more of second-guessing type than a book for self-study: it has cryptically written sections, no examples (and those given, and rarely, are even more ...
10
votes
2answers
257 views

Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between ...