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.)

Section 6.2 (part 2) due 10/15

I don't fully understand the terms "natural homomorphism" or "homomorphic image." It says that if S is a homomorphic image of R then R and S might have the same algebraic properties.

It seems that this section is important because it relates things that can be helpful when writing proofs. It is interesting that the section says, "A concept developed with one idea in mind may have unexpected linkages with other important mathematical concepts. That is precisely the situation here." It's neat that there are other results than just what they had wanted.

Section 6.2 due 10/13

More than any other section in the book, I am having a hard time visualizing this (as much as Abstract Algebra can be visualized). In Theorem 6.8, are a, b, c, and d representing numbers? All of this new notation, including and especially "R/I", will take some getting used to.

If I understand this section accurately, then this is the method behind doing multiplication in my head. For example, if I={x in R 3y=x} and a=2, then 83=a+nx=2+(27)(3). Is this what they are saying?

Section 5.2 due 10/6

I don't understand it Theorem 5.7 when it talks about F* being an isomorphic subring of F[x]/(p(x)). How is F* different from F?

This is pretty straight forward because a lot of theorems about rings are now applied to polynomials, like the rules of modular arithmetic. However, there are some new theorems that, to me, don't seem as obvious as others, but will probably be helpful in proofs.

Sections 4.5 and 4.6 due 10/1

Theorem 4.29 says that if a polynomial in the reals is irreducible then it is either of degree one or two. In the proof of it, it says that this is not necessarily the case--that higher degrees have roots in the complex numbers. Because of the way it is stated in the theorem, though, it makes me wonder if I am understanding it correctly. Are there really only irreducible polynomials of degrees 1 and 2?

There are many theorems in 4.5 about primes, coefficients, etc. Some of them seem obvious while others seem brilliant. All of them make me wonder if they are applicable. (I suppose they are because they are in the textbook.)