Section 7.2

Difficult: The hardest part of this section for me was when the author started to talk about the order of an element. I mostly understand it, but it was a little difficult to follow the proof for Theorem 7.8. I always have to go back to the definition of finite order to understand what the theorems are talking about. Because I had a hard time understanding Theorem 7.8, Corollary 7.9 was hard to follow as well. I think seeing more examples of finite order would help me.

Interesting: I am now seeing that basic properties of groups have been included in many of my previous classes. For example, in linear algebra, we talked about the inverse of two matrices (AB) being equal to "B inverse" times "A inverse." Now it makes sense and I can see the connection to a real example. It is also interesting that many of the things that we assumed in calculus and other math classes are now being defined and proved. I can compute and operate in certain ways because the number systems or algebraic systems that we work with are groups, rings, and fields.

Section 7.1, second part

Difficult: At first, it was hard for me to accept that any movement of a square would result in one of the eight permutations. Then, I realized that the authors were quite specific when describing what the result of the movement had to be. I now understand that the square has to be in one of eight positions after doing a permutation on it. Similarly, other examples in the text were difficult to grasp at first, but after thinking about them further, I was able to understand how they qualified to be groups.

Reflective: I think that the concept of a group is interesting! We can even classify groups in terms of symmetries of shapes. I can see how these ideas are applied outside of "pure mathematics." I wonder, though--how can knowing about groups help us prove things in the future? I just want to know how group theory fits into the big mathematical picture.

Section 7.1, first part

Difficult: The most difficult part of the text for me was, in fact, the definition of a group. I need to go through the different parts of the definition over again so that I can review and understand it better. It was hard to understand because so many of the things we define in this class have a list of properties/characteristics. It is starting to become blurred in my mind. I think it would be helpful for me to see more examples of the difference between a group and a ring.

Reflective: We talked a little bit about permutations and groups (without calling them groups) in my Math 290 class. It is interesting to me that the set of permutations (the example in the book) is a group! I like to think about it because it isn't calculus or difficult crunching of numbers. It is a simple example that shows that mathematics is everywhere. It seems to me that I will find a lot more groups because there are less qualifications required to be a group. I'm excited to see more examples.

Section 9.4

Difficult: At first, the most difficult part of the section was getting accustomed to the new notation and concept of defining "fractions." However, it was fairly easy to understand once I got used to it. Lemma 9.29 and Thm. 9.31 were more difficult for me to understand. I followed the proofs, but I don't see how they work. I would like to see an example of these two theorems. I also didn't understand Thm. 9.31 because the proof seemed inadequate. I did not understand how the proof proceeds.

Interesting: We talked a little bit about the field of quotients in my Math 290 class (of course without calling it the "field of quotients"). It is interesting to me that we can explain how division and fractions work by defining an equivalence relation. I like making such a simple concept extend to other rings. I want to look for other integral domains and see what "fractions" look like in these rings. Hopefully we will cover some examples in class.

Preparing for Exam 2

1. Important topics and theorems
I think that the most important theorems and topics for the exam are the basic theorems about ideals. These seem to encompass everything we have talked about so far. I also believe that the theorems about F[x]/(p(x)), like Thms. 5.7 - 5.9, 5.11, and (of course) the theorems that you have mentioned--like the "named" theorems in the book, as well as the theorems we need to know well.

2. Better understanding
I need to work on better understanding chapters four and five. Although chapter six is difficult for me, I am starting to grasp it better. It was really hard for me to understand and comprehend the structure and the theorems regarding F[x]. I think we talked about it abstractly, so it was hard for me to "picture." I have been studying and really trying to understand the fundamentals about the polynomial rings. Hopefully my studying will pay off and I will be able to grasp the information and the characteristics. F[x] mod p(x) is extremely difficult for me to understand. I think that "modding out" by p(x) still doesn't make sense to me.

3. Wednesday class
I would like to do a homework problem that we haven't done before from sections 4.3 or 4.4--maybe a problem about proving irreducibility from sec. 4.3 and numbers 15 or 18 in section 4.4. I would also like to see a few more examples about the congruence classes/elements of F[x]/(p(x)). Visualizing it would help me a lot.

Section 6.3

Difficult: The hardest part of this section for me to understand was exactly how to determine that a quotient ring is a field. I understand how to prove that something is a field in general, but in the book, the author assumes that if a ring is isomorphic to a field, it is also a field. Did we come to this conclusion before? We probably did, but it was confusing to me while I read because it was stated without any reasoning. It would help me to see and talk about some examples in class.

Reflective: Primes have always interested me, and I think that it is really interesting that we can extend the idea of a prime to ideals. It makes sense, and I think it is amazing that prime numbers have so many applications. I want to know more, though. Why does understanding about prime ideals help us? I know it must have some important application or reason, but right now we are just learning it. I wonder what prime ideals tell us.

Remaining Section 6.2

Difficult: The hardest part of this section for me to understand was the first isomorphism theorem. I was struggling to follow all of the new terminology before, but when I got to this theorem, I got lost. I think the hard part for me is not being able to "visualize" the theorem or the concepts discussed in 6.2. It would be helpful for me to see examples of the theorems in order to understand more fully what they mean. The idea of a "quotient ring" is also difficult for me to comprehend, partly because I still struggle with the idea of taking a ring R modulo I.

Reflective: Although this section is really abstract and hard for me to understand, it is bringing back ideas from Linear Algebra. The idea of a kernel makes more sense to me now, and it is interesting to think about the influence of an isomorphism on two rings. It is really amazing that by constructing an isomorphism between a quotient ring and a homomorphic image of a ring, we can understand a little bit better about the structure of quotient rings, especially ones that we have seen before.

Section 6.1, part of 6.2

Difficult: The hardest parts of these sections for me were the concepts of principal ideals, ideals generated by an element c, and the concept of R/I in section 6.2. I think I understand what an ideal is, but it is hard for me to grasp all of the characteristics of ideals without really spending time with examples of them. The notation of a + I is also a bit confusing, but as I kept reading, I grasped it better. Another difficult part of the text for me was the fact that the authors simply stated that the proof of each theorem was the proof of a previous theorem with the theorem numbers switched and a notational change. I would like to see a couple of proofs to wrap my mind around the connection between ideals I have already seen and other ideals in general.

Reflective: The most interesting part of the text for me was the concept of an ideal itself. It was interesting to see that there are many "nice" properties of ideals. However, I still don't know why we are learning about ideals. I don't know how they help us to understand the algebraic structure better. I believe that I will discover this as time goes on. It is interesting that everything we study in mathematics always has a deeper meaning and reason for existing. Mathematics never ends; we can always go deeper and learn more.

Section 5.3

Difficult: The most difficult part of the text for me was understanding the difference between F[x], F[x]/(p(x)), K, and F. All of these fields are now blurring a little bit and so the concept of an "extended field" was a little hard for me to comprehend. However, after spending quite a bit of time reading, I feel a little more comfortable with it. It is also a little confusing that the extended field K of F contains roots of polynomials in F[x] which do not have roots in F[x] itself. I don't quite understand what that all means.

Reflective: By the end of the section, I was beginning to understand the concept of the extended field K a little more. I found it amazingly interesting that the field C is actually just R[x]/((x^2 + 1)). Now the fact that there exists a number i makes sense! It is helping me to understand the existence of the complex numbers. Now I understand why my teachers in high school and at BYU focus on the complex numbers so much. K is more comprehensible because I can associate it with the field C.

Section 5.2

Difficult: The most difficult part of the section for me to understand is the concept that F[x]/(p(x)) contains F. It is hard to comprehend this fact, but if we prove it in class, I think that I will understand better. Examples would also be helpful to me. The last theorem was harder for me as well--but I think it was hard because I still struggle with the idea of a unit in a ring. Whenever I prove things about units or associates or consider a theorem involving them, I still have to go back and look at the definitions. I just have a hard time understanding what they mean.

Reflective: The most interesting part of the section was also the hardest for me to understand. While it is a little abstract that F[x]/(p(x)) contains F, it is a really interesting idea. I still need to understand it further, but it is interesting to consider. I think that the polynomials are becoming more interesting to me because I am discovering more about them.

Section 5.1

Difficult: Although the theorems were fairly simple and I had seen them before in chapter two, I still find it difficult to directly apply them to F[x]. In this section, there were hardly any proofs, so it was hard for me to visualize the theorems in F[x]. Hopefully we will go over some of the proofs in class so that I can understand them better.

Reflective: I think that the theorems about congruence modulo n in F[x] are really interesting. I never thought of applying rules about mod n to F[x]. I want to investigate further and find out if congruence mod n can be recognized or is useful in other contexts. I hope to look at this more.

Assignment 14: Sections 4.5 - 4.6

Difficult: The most difficult part of these sections for me was understanding the proofs presented in the text. The two most difficult theorems for me to understand were the theorem about the irreducibility of a polynomial in Q[x] based on its irreducibility in Z[x] and Einstein's Criterion. It is interesting that they work to find which polynomials are irreducible, but I don't quite understand why.

Reflective: The most interesting section for me was section 4.6. I think the fundamental theorem of algebra is really exciting. I like C[x] and other rings or problems dealing with complex numbers. I think it is very useful that every polynomial in C[x] (except polynomials of degree one) is reducible. This theorem is easy to understand and interesting to consider. Imaginary numbers really do explain a lot of characteristics of polynomials.