Questions, Due April 11, 2012

Honestly, I feel that all the theorems that we need to know for the final are the most important. :) But written below is a theorem that I feel is important, but I am not exactly sure why. I can tell that if I understand this theorem better than a lot of other things will make much more sense to me, like quotient rings and groups for example. Anything dealing with those I feel is just very difficult for me for some reason, I do not know why.

• Theorem: If R is a commutative ring with identity and I is an ideal of R, then R/I is an integral domain if and only if I is a prime ideal. 

The general topics of the different kinds of rings and groups I think would be good topics to know. 

Any past homework question relating to A(n) is something that I need to understand how to do because I am not even sure what A(n) is. How do I even go about finding it? 

If there is one thing that this course has taught me I think it would be the importance of being exact, precise, and disciplined. There is great power that comes from exactness, and I think I am just starting to understand it, because it is a hard thing to learn how to always do. Also I have learned better how to problem solve and look at problem from the inside out, step by step, instead of the whole mountain at once. And of course this class has taught me patience. Patience in myself, my work and in those around me.  

Sections 8.4-8.5, Due April 5, 2012

Difficult:

As I read section 8.4, I had difficulty distinguishing between C and C(a). I know that C(a) is the centralizer of a. But then they used C in place of C(a) and then they used it as the representation of the conjugacy class and I got a little confused. Also, with how intricate all of the details are becoming about all the theorems and what we can and cannot say about any given group, it is difficult for me to really understand the why behind all of the theorems, especially those in 8.5, because to me many of them seem somewhat trivial. It seems random, even though I know it is not. I know that there are a lot of good reasons why someone discovered these theorems but, like I said, a couple of more examples would be quite help.

Reflective:

It is a sad thought, but I think I am just now starting to really appreciate abstract algebra better. It really is a fine, and complicated art. What is so funny about it is that there are times when the book seems to be making things so very simple and easy. And then all of a sudden it comes to a theorem and its proof and all simplicity gets thrown out the window at times. I just hope that I can study enough so that I can look at the final from the simplest view possible rather than trying to cram everything in and only make it unnecessarily complicated for myself.

8.3-8.4 Sections, Due April 3, 2012

Difficult:

I understand what a Sylow group is, the difficult part is understanding how it can be used and all of the different proofs about it. Any proof or theorem involving the group "K" for some reason to me is confusing. I do not see it as plainly as it seems like it is supposed to be seen from the theorems. Theorem 8.17 is particularly confusing to me.

Reflective:

I am really looking forward to the day in this class when we will stop moving forward. I know that that may sound foolish , but I honestly only just want to stop and give myself time to understand what we have already learned. Just when I think that I am about to understand things, we are introduced to a brand new kind of way to organize letters and groups. It can be very frustrating.

Section 8.2, Due April 1, 2012

Difficult:

I do not completely understand the concept of G(p), therefore it was difficult for me to understand the proof of Lemma 8.4. The paragraph at the top of page 255 is also confusing to me. I do not understand exactly what a p-group is or an element of masimal order. Also, even though the book said that I should be able to follow the logic of the proof of lemma 8.6, it was difficult for me to follow. I still do not understand it.

Reflective:

What we are now learning about groups has gotten to be very detailed that it is difficult for me to follow most ideas and concepts. I do like groups better than rings for some reason. I think it is because I think that they are more visual. A cyclic group, for example, is a group that I can visualize very well. So, hopefully as I do problems relating to this section I can better visualize what is happening.