Assignment 13: Section 4.4

Difficult: The most difficult part of this section for me was making the connection between the roots of a polynomial and the irreducibility of a polynomial. It conceptually makes sense, but I need to think about it more and analyze the proofs in class in order to understand it better. I also at first had a hard time distinguishing between f(x) as an element of F[x] and f(x) as a rule for a polynomial function. Although these concepts are difficult for me now, I know that discussing them in class will help.

Reflective: I think learning about the characteristics of polynomials in terms of the ring F[x] is helping me to understand the properties I learned in middle and high school. I am beginning to understand polynomials on a deeper level. I think this will help me to understand what a "root" is below the surface level.

Assignment 12: 4.2 and 4.3

Difficult: These sections were already familiar to me because we had already proved similar facts about elements of Z. However, the book does not prove many of the theorems, and it is hard for me to just apply different definitions that I do not fully understand yet. I think it would be more helpful for me to understand if the book supplied more proofs instead of simply stating "adapt Theorem ___." Overall, the concepts are not too difficult.

Reflective: The sections deal with polynomials; I had never made the connection between polynomials and elements of Z. I know most of the characteristics of F[x], but it is still hard to prove them. However, knowing this will help me to teach the properties of F[x] to my future students.

Assignment 11: Section 4.1

Difficult: This section, in general, was very difficult for me to understand. I could follow most of it, but since there were few examples, I couldn't picture what was happening. Thus, my understanding of F[x] is still greatly limited. I think I need to explore simple examples to understand the proofs. The hardest part was the proof of the Division Algorithm for polynomials in R[x]. I don't understand why it works. Hopefully we review it better in class.

Reflective: This new concept of polynomials with coefficients in R and x= some element in a greater ring P but not in R is new and intriguing. I don't understand why we are studying it, but I hope that more understanding comes with time. It is interesting to think about what x actually means. I imagine that this "mystery" about x is confusing to many students when they first learn algebra and symbolic manipulation.

Assignment 10: Preparing for Exam 1

Most Important Topics and Theorems: I think that among the most important topics we have learned are the division algorithm, the greatest common divisor of two numbers, properties of prime numbers, modular arithmetic, congruence classes, properties of Z(n) and Z(p) when p is prime, rings (their definitions and properties), and isomorphisms and homomorphisms. Some of the most important theorems are the fundamental theorem of arithmetic, the division algorithm, theorem 2.8, theorem 2.7, Corollary 2.5, theorem 2.1, theorem 1.8, and theorem 1.5.

Expected Questions: I expect to see questions about definitions on the exam. I also expect to have to prove major theorems we have learned in class. I know that we will have to prove other theorems using the definitions and theorems that we have studied, but it is hard to know what theorems to focus on. I want to thoroughly understand all of them, but I know that some are more important than others when considering proving new theorems. I expect the questions to be focused on the definitions and theorems. I expect that if I know the definitions really well, I will be able to prove the theorems.

Assignment 9: Section 3.3

Difficult: The most difficult part of the text for me was verifying if two sets are isomorphic and homomorfic. The concept is fairly new. In my Math 190 class, we spent considerable time proving if functions were injective, surjective, and bijective. This was rigorous for me, but now we have to prove that a function is bijective and prove part (iii) of the theorem in order to show that the two sets are isomorphic. I think I just need practice doing so.

Reflective: The most interesting part of the section for me was the very beginning. I loved how the textbook showed the literal "replacing" of elements of Z(5) with elements of the new set S. This illustrated really well to me what it means to be a bijective function and what it means for two sets to be isomorphic. I want to use examples like this in class when I teach so that I can help students "get it."

Assignment 8: 3.2

Difficult: The most difficult part of this section for me was trying to conceptualize all of the various properties of rings. I know that the theorems are quite familiar to me with rings like R, Z, and others, but I tried to think "outside the box." Instead of familiar elements of R, Z, Q, etc., I tried to think of a, b, and other elements used as elements of any ring. This confused me sometimes. It is hard to remember what theorems and axioms we can use to prove other theorems and corollaries.

Reflective: If I can understand the properties of rings, I think I will be a better math teacher. I will be better equipped to help students to understand why and how math works. Understanding rings in general will help me to understand well-known rings on a deeper level. In addition, as we move to performing algebraic actions on elements in rings, I think algebra will get harder! I will be able to understand my students better when they "can't understand" algebra because it is "symbolic" or "abstract." I am excited to learn more so that I can connect with my future students.

Assignment 6: Section 3.1

Difficult: I think the hardest part of the section for me is just learning how to recognize a ring and determine when a set is a ring or not. I understand the axioms describing the characteristics of rings, but I just need to be a little more familiar with them. Another question that keeps creeping into my head is why? Why do we study rings? How does it help our understanding?

Reflective: Rings are interesting sets; I want to learn more about them. I get the feeling that by learning about rings, I will understand R, Q, C, and other sets better. I want to understand the numbers so that I can effectively teach mathematics to my future students.

Assignment 5: Section 2.3

Difficult: The main concepts of this section were fairly easy for me to understand. However, I could not quite understand the proof for Corollary 2.9. I think I may have missed an important step because I stop understanding when the authors begin to prove the uniqueness of the solution of ax = b in Z(p). I would like to go over the proof in class.

Reflective: Like I stated at the beginning of the year, I am quite fascinated with prime numbers! I think it is so interesting that Z(p) when p is a prime can have so many interesting and convenient properties. I would like to study prime numbers more. I think I need more practice finding the unique solution for ax = b in Z(p) or in Z(n) when (a, n) = 1. I understand it when I read it, but my brain doesn't quite grasp all of the reasoning or the application. I would like to go on and study more about this set of prime numbers. I hope we keep studying it!

Assignment 4: Section 2.2

Difficult: Modular arithmetic is hard to get used to, but I am quite familiar with it now. When I first read Theorem 2.6, I did not understand why this theorem proved that the definition of addition and multiplication for congruence classes is independent of the integers that are chosen to compute. I kept reading, but after a few minutes, I returned to the section and read the theorem again. I finally realized that the hypothesis was the key. It proved the definitions of addition and multiplication because it is supposed that [a] = [b] and [c] = [d]. I now understand why we can assume that [a] + [c] = [a + c] and that [a][c] = [a*c].

Reflective: The part of the chapter that is most interesting to me is the part that reviews the addition and multiplication tables for different values of Z(n). I like doing modular arithmetic because it helps me to better understand why we use the methods we do in "normal" arithmetic. I better understand our number system, normally using base 10, because I see the relationship between the "remainder" and the number represented. It is easy to make mistakes when doing modular arithmetic, but I enjoy it.

Assignment 3: Section 2.1

Difficult: I read this section carefully, trying to capture each idea. I have learned these concepts before, but they were extremely difficult for me the first time. It is refreshing to learn them again! I think the hardest part of the material for me to understand is the meaning of modular arithmetic. After taking a few classes that have used it, I have become much more comfortable with it. However, I want to understand why we do modular arithmetic--what is advantageous about modular arithmetic? If I understand modular arithmetic, how will this help me be a better math teacher? How will it allow me to access further mathematical knowledge? Is this a key element in the foundation of abstract algebra?

Reflective: For me, the most interesting parts of this chapter were the theorems about congruence classes and their relationship to the set Z(n) and congruence mod n. It is interesting to me that there are n congruence classes from [0] to [n - 1]. When we start to experiment and use modular arithmetic, the fact that there are n congruence classes starts to make a lot of sense. I want to better understand the congruence classes because I think it will help me to better understand numbers and their relationship to each other. I want to be able to recognize the relationships between numbers more quickly. Understanding modular arithmetic and congruence classes will help me to do so.

Assignment 2: Sections 1.1 - 1.3

Difficult: The most difficult part of the material for me was the concept behind the Theorem involving d = au + bv. I have learned this theorem before (as well as the others in the chapter), but the concept is still hard for me to grasp. I know that it is true, and I know how to prove it, but I struggle understanding why it is true in my brain. I need to think about it more and let it sit in my head to understand it better. The other part of the section that was difficult for me was the amount of theorems in it. The proofs were easy to follow, but I hope to be able to remember all of the theorems and all of the proofs.

Reflective: I am fascinated by prime numbers. I have been interested in them for a while. Section 1.3 was very interesting to me because it laid out various characteristics of prime numbers. I think it is interesting to note that the Fundamental Theorem of Arithmetic is the fact that integers are products of prime numbers. I don't know why, but the theorems regarding prime numbers are profound to me. I think that the fact that prime numbers have special features and characteristics is very interesting, and every time I read about prime numbers I want to learn more about them. I like understanding the characteristics and character of the number systems so that I can better explain them and help other people to understand them as well.