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