Section 7.3 due Nov 1

I didn't understand the proof of Theorem 7.11. Many of the other theorems are obvious.

As in rings, it is helpful to know that G is a group so that you don't have to prove as many things, like associativity in subgroups.

Section 7.2 due 10/29

It says a^n=aaa...a (with n factors) but (ab)^n does not necessarily imply (a^n)(b^n). So does (ab)^n=(ab)(ab)(ab)(ab)...(ab) (with n factors of ab)? They said that we will use standard multiplicative notation, so shouldn't that mean that this is commutative?

The proof of part one of 7.5 was at first confusing but, upon further review, made a lot of sense and seemed extraordinarily creative. Then again, when are proofs not creative?

Section 7.1 (part 2) due 10/27

The section says that a nonzero ring is never a group because of the zero ring. Since multiplication is such a straightforward operation, why do they not make an exception in the group to say that for everything else, the group will work?

I thought the reflections were interesting because I haven't thought about reflections of shapes for a long time. I also think the theorems were somewhat interesting because they find some groups for us.

Section 7.1 (part 1) due Oct 25

With rings they say that if ab=ba then the ring is commutative. On p 163 they say that a group is abelian if it satisfies the commutativity axiom. Why is there a special word for it? Do they do it since multiplication commutativity is much more simple than what some operations can be? Can all groups' permuations be represented by an array?

Function composition is associative, but how often are composed functions commutative? Do they have to be simple, linear functions? That would be something interesting to explore a little bit. (Of course, they must not be too rare because there is a special name just for this type of group.)

Midterm course evaluation

1. The homework and the blogs contribute most to my learning. I like the blogs because it forces me to read the section before we talk about it in class. I think it helps you teach better because then you know what to spend more time talking about. The homework helps because it really is challenging and gives us a good idea of what kinds of questions might be on the test.

2. In class, it seems like we go over the proofs really fast. I feel like I am so busy taking notes that I am not taking everything in. (This doesn't seem to be true for the majority of the class, though, since they ask lots of questions.)

Review for test due Oct 20

I think ideals and quotient rings (and basically all of chapter 6) will be the most important topics that we have studied that will be on this test.

Ironically, I feel most uneasy about quotient rings. It seems like there is a lot to know about them (even though it is really only covered in one section) or at least a lot that can be proven from them, thus, it is necessary to study this more intensely for the test.

The question I would like worked out in class comes from 6.3 #11: Show that the principal ideal (x-1) in Z[x] is prime but not maximal.

Section 6.3 due 10/18

I don't understand the concept of a maximal ideal. And why can there be multiple maximals?

It is neat that all the concepts we have learned are interrelated and all the theorems relate these concepts. (Unfortunately, I don't understand some of these individual concepts so I can't yet see the bigger picture for 6.3.)