# 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 applications. I would like to get to the impossibility of squaring the cube, trisecting an arbitrary angle with a ruler and compass, etc but I don't think that there will be time. (If you disagree with this attitude, please point me in the direction of an appropriate text book that gets from nothing to minimal polynomials in less than 20 weeks.)

I am therefore brain-storming possible two-term abstract algebra classes, ideally containing both theory and applications (to mathematics or to other disciplines), and ideally being more deep than broad (the second term should use the first). Ideas I have had so far include:

1. try to get to groups, rings, and fields, at the probable cost of ending before we've arrived at any rewarding application
2. all group theory: the basics (homomorphisms, quotients, etc), then study group actions enough to solve the Rubik's cube or prove Sylow's theorems.
3. all ring & field theory: homomorphisms, quotients, prime and maximal ideals, polynomial rings, factorization, and extension fields

Please note that the students taking this class will be mostly working on secondary education degrees, and relatively unlikely to be planning to study mathematics in grad school. This discourages me from suggesting something like group theory 1st quarter, intro to representations in the 2nd quarter. (Feel free to disagree with me here, too)

• What kind of background do the students have? What are the prerequisites? It's going to probably be very different if you are teaching to students who have already had a rigorous proof based math class (mathematical analysis as opposed to calculus in the US). If this will be one of the first classes I think it's a great subject to really focus on getting into mathematical reasoning. Algebra can easily be taught as very abstract and thus gives students a chance to actually learn some mathematics rather than just more arithmetic which is what most of lower level US courses are. – DRF Aug 31 '17 at 6:51
• @DRF They will have had an intro to proof course, two quarters of linear algebra, and a year of calculus, at a minimum. – David Steinberg Aug 31 '17 at 17:30
• The Diffie-Hellman key exchange algorithm is interesting. It is easily explained and is a nice application of cyclic groups (or subgroups). Its original version was in terms of the multiplicative group of nonzero integers mod $p$ (where $p$ is prime), but has been extended to other groups as well, notably elliptic curve groups. See this: en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange – John Coleman Aug 31 '17 at 18:37
• You should definitely look at Rotman's undergraduate text. I'm not sure about timing, but, it is wonderfully written and has some really careful arguments for basic group theory. – James S. Cook Sep 2 '17 at 2:49

Abelian group theory emphasis with application being point symmetry (playing cards, stars, polyhedra, buildings, molecules, etc.. I wouldn't go into IR stretching modes, but you could at least mention that point symmetry is very practical and important to chemists and biochemists doing spectroscopy.

I would try to really take it easy with the abstractness and the rigor but give them some feeling for why they should care about it. None of them will teach it in high school so it is not so important that they are proofed to death. But if they at least know the story (even without the details) that group theory helped show the quantic is not solvable, etc. That will inform them about what it is about. And can come out in HS algebra class as a remark.

• It is also worth mentioning here as an aside that affine transformations play a significant role in Common Core geometry, and in some curricula high school geometry students are expected to realize that, for example, doing two reflections in a row can result in a translation or a rotation, depending on whether the lines of reflection cross or not. The representation of affine transformations in the plane as a group represented by matrices is concrete, and applicable to future geometry teachers. (I taught HS geometry two years and am moving on to a math PhD.) – Opal E Aug 31 '17 at 17:55

Given your clientele, I think focusing mostly on groups and very concrete rings is good. Or skip more advanced group theory to do the trisection, though perhaps you are right that you just need too much group theory to get there.

However, my own idea is that I don't see why you couldn't do quite a bit of number theory (I will admit I am biased) from the groups standpoint. It fits in very nicely. Likewise things like frieze/crystallographic groups. So basic groups, a bit of rings, then lots of topics. Not as comprehensive, but probably better in the long run - ?

In general, there are several textbooks out now addressing these groups specifically ... here are some random links.

I would like to recommend you at least look at Nathan Carter's Visualizing Group Theory (MAA link), even if you don't use it as a textbook (although I think it would make a fine textbook, especially for education-degree students). You could supplement with exploring crystallographic groups, as kcrisman suggests, perhaps culminating in Conway's "Magic Theorem" (discussed, e.g., in The Symmetries of Things). This could easily fill two quarters.

• I enthusiastically recommend this as well, though not as a standalone text. Don't forget groupexplorer.sourceforge.net (not sure if it still works on newest Macs); unfortunately the little App store game he made doesn't seem to currently be available. – kcrisman Aug 31 '17 at 1:41

As someone who learned algebra from the old classic by Herstein-probably too difficult for your students, but a jewel worth browsing by any instructor-I think it's important to cover group theory as thoroughly as possible The reason for this is that subsequent algebraic structures-rings, fields,modules-are all most directly constructed by adding structures to a group. For example,a vector space can most easily be visualized as a structure where the Abelian group of scalars in a field acts on an Abelian group who's elements we call vectors. Another example is how the structure theorem for finitely generated modules over a principal ideal domain is a straightforward generalization of the classification theorem of finite Abelian groups. I would cover as much group theory as possible and then build ring theory on top of it. Not only will it go much faster since most of the basic results will be familiar from group theory,it will be much pedagogically clearer to the students.

I would also try and add as much geometric content and applications as I can at this level. The use of symmetry groups and the decomposition of modular arithmetic gives 2 very important applications using group and ring theory.

Good luck!