4.5, due on June 1

(difficult)
The two or three proofs after Lagrange's theorem were difficult.

(reflective)
So groups now have the congruence relation. It will be interesting to see what will be the next level of abstraction.

7.3-7.4, due on May 27

(difficult)
Homomorphisms, isomorphisms were the most difficult parts for me. Switching back and forth between additive and multiplicative groups. Those funny groups formed by
(reflective)
It's interesting to see the tie in from rings to groups. There was the discussion in class of the optimal way to approach the course. I wonder if, in other books, rings are treated like special groups. Obviously, rings require additional and more complex axioms. But perhaps group theory is richer, and so deserves to be treated second?

6.1, due on May 18

(difficult)
All very similar theorems and definitions, so maybe the intuition is the hard part. Some examples would probably be helpful. Also, if we have time, more talk on 5.3 would be welcome.

(reflective)
Generalizations are nice. It will be interesting to see where this goes.

5.1-5.3, due on May 16

(difficult)
The author stated several times the grand significance and subtlety of the results of these sections; I struggled to understand the implications. Some help flushing them out would be nice. It seems the example of the equivalence between the complex numbers and R/(x^2 + 1) is instructive.

(reflective)
Again, the author said these results are profound, and I'd like to appreciate more fully what he intends. The definition he gives for the complex numbers is intriguing.

4.3-4.4, due on May 13

(difficult)
The proof of theorem 4.11 seemed difficult (even though it's analogous to a theorem in ch 1). It seems like the difficulty of the proofs increased in this section.

(reflective)
I thought the method of dividing f by (x-a) to find f(a) (in 4.4) was very interesting. I suppose we may use this property again when we talk about more general rings.

Apdx G, 4.1-4.2, due May 11

(difficult)
It would be useful to work through the definition of multiplication in R[x]. I've had a hard time with it before.

(reflective)
The author talks about x like we've been using it incorrectly this whole time. I suppose this viewpoint indicates a way in which algebraists view mathematics differently from other mathematicians. Or perhaps I don't understand how other mathematicians view it.

3.3, due May 9

(Difficult)
It seems like the proofs at the end of the section will be the most difficult, conceptually. These are finding a homomorphism, proving there is an isomorphism, proving two rings are not isomorphic, etc.

(reflective)
I can never seem to remember what an iso- or homomorphism is. I'm happy to see it another time. Also, proving a function is one-to-one and onto is very useful

3.1-3.2, due on May 6

(Difficult)
Another long section with a lot of theorems. Due to the section's rigor, the proofs all relied on the axioms for commutative rings with identity, integral domains, and fields. The most difficult part of this section will probably be the need to internalize the axioms so they can be used offhand.

(Reflective)
While rigor is difficult, it's a great exercise in developing a system from the ground up. It will be interesting and useful to be along for the ride.

2.3 3.1, due on May 4

(Difficult)
Learning the definitions of 3.1 seems like it will be the most difficult. I suppose it will be necessary to memorize the various axioms that must be satisfied for each type of ring or field.

(Reflective)
3.1 begins the sections on abstract algebra. The introduction to the chapter was interesting--Learning how to group mathematical systems by their properties will be a useful exercise, and gleaning what those properties imply will sharpen my ability cut to the essence of any problem.

2.1-2.2, due on May 3

(Difficult)
From section 2.2, the most difficult topic was addition and multiplication in congruence mod n. While I understood it, the intuition will be hard to internalize.

(Reflective)
It will be interesting to see how these equivalence classes will be used like vector spaces.

Introduction, Due on 28 April

I'm a senior (graduating after spring term), majoring in econ and math.

I've taken the following math classes: linear algebra, ord. diff. eq., multivariable calc, 290, real analysis 1&2, numerical analysis, complex analysis, and topology. I'm taking this class to finish the major.

My most effective math professor was Dr. Lawlor, for real analysis 1. It was my first real proofs class, and he effectively reduced the difficult concepts to their essence. His exposition was clear, and the logical progression was obvious.
My least effective math professor was Dr. Swenson, for complex analysis. He breezed through the proofs, and seemed to assume the class would be able to sort it all out. If I had not already taken real analysis 1&2, I would have been lost.
I guess the moral of these two cases is to find a happy medium. While I learned much with Dr. Lawlor, he often coddled us. And although Dr. Swenson rushed through the proofs, he treated us like math majors (which, I feel, is the level at which a 300 level class should be taught).

Next fall, I will start the econ Ph.D. program at the University of Rochester. Hence, I will probably slack off in this class, because grades don't really matter at this point. I'll still participate in class and do the reading and homework and all that jazz, though.