Section 4.2, Due Wednesday Feb. 1

Part I: What was the most difficult part of the material for you?

The proof to Theorem 4.5 was the hardest thing for me to follow. It is difficult to read and follow the proof all the way through to show that d(x) is the unique gcd. I understand how there is a gcd but it seems more difficult to see that it is the only one.

Part II: Write something reflecting about the reading.

It is interesting to compare these theorems to chapter 1 and see the similarities. I never would have expected polynomials to act so similar to integers. I am just guessing that the next section will continue on into showing how mod works with polynomials as well. 
 

Section 4.1, Due on January 30

Part I: What was the most difficult part of the material for you?

I am most confused on what the F[x] represents and if f(x), g(x) is an element of F[x]. Does that mean g(x) and f(x) are the coefficients for the polynomial or does it mean f(x) and g(x) are the coefficients with the x part, so the whole term? This is what was mentioned in Theorem 4.4 The Division Algorithm in F[x] and I just don't fully understand the concept of F[x] yet.

Part II: Write something reflective about the reading.

 I really like seeing how the division algorithm works and the proof they use in the book. It was really helpful to see an example next to the proof so you could understand what they were meaning better. Even thought I was confused with the meaning of Theorem 4.4, I think I can understand the computational side of how to do it which I like.

Other Questions, Due January 27

How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

I spend at least 2 hours on each homework assignment. I try to work with other classmates whenever I can and I go to the T.A.'s office hours so I can understand it the best. The lectures and the readings help me prepare for the homework, but there are still some questions on the homework I may be completely lost on even though I did the reading and went to class.

What has contributed most to your learning in this class thus far?

Working with groups on the homework has helped me the most. But I also think that reading the section before the lecture has been helpful once I go to class because most of my questions will be answered.

What do you think would help you learn most effectively or make the class better for you?

There are times when I have worked hard on the homework and I still have a few questions, so if time allows for it, it would be nice to go over a question on the homework at the  beginning of class.

Section 3.3, Due on Januray 25

Part I: What was the most difficult part of the material to you?

 The most difficult part of the reading for me was the statement and proof for Corollary 3.13. I am really confused about the idea of the image and how it relates to S. I understand that the image is f(r) for any r that is from R, but it is hard for to me relate this idea in the proof since I is just stated to be the image, when I could say that I want to see f(r) to be the subset of S, right?

Part II: Write something reflective about the reading. 

I really liked the example in the book about how to find a function that is an isomorphism and it had integers mod 12 and the cross product of integers mod 3 and 4. It built up so well to each step and to the final product, yet it seems super complex because there are so many subscripts and numbers with functions, cross products, and integers. It could be easy to get lost but you just have to look at each step carefully and it makes sense.

Section 3.2, Due on January 23

Part I: What was the most difficult part of the material to you?

The most difficult part was understanding the proof with theorem 3.6. It was hard for me too see the connections from this theorem to Theorem 3.2 and I think if the book had made it more organized rather than used numbers to relate back to statements, I would understand it better. Hopefully in class it will make more sense as to how only those 2 statements in Theorem 3.6 are needed to prove a subring rather that the 4 statements in Theorem 3.2.

Part II: Write something reflective about the reading.

I really like the definition at the end of this section about what a zero divisor is because the real numbers and the integers do not have a zero divisor but the integers mod six have one  and so many others. For most of this section, it kept showing how the integers and rings have so many common properties, but at the end they were different properties. Connecting these properties from the integers into other rings seem too easy at first because these properties are so second nature, but the zero divisor is easier to see the importance because it isn't in the integer.

Section 3.1, Due on January 20

Part I: What was the most difficult part of the material for you?

The first example that was mentioned was confusing for me to follow just because they mentioned that the product of the integers mod 6 crossed with the integers is defined in Appendix B but I couldn't find it. I eventually moved on from finding what it meant and I still understood the problem but I didn't understand with what it meant by integers mod 6 crossed with the integers.

Part II: Write something reflective about the reading.

I think it is interesting that the axioms needed to prove a subring are so much shorter than to prove if something is a ring. I feel like there is more to think about in this theorem so that I can make more sense in why we must only prove those 4 axioms, but I am willing to believe because of the examples that were done for now. But with the fourth axioms for a subring, I need to think more about that because it seems like we should find if solutions exist but apparently we already know they exist.

Section 3.1 through middle of page 48, due on January 18

Part I: What was the most difficult part of the material for you?

The most difficult part of the material for me is trying to match up all the definitions with certain groups of numbers. Once the definition of a field came up, I figured that, for example, since all real numbers are a field, that would mean they satisfy axioms 1-12 but then I noticed that the definition of a field doesn't mention Axiom 11 but apparently it does hold in fields. That is when I get a little confused because it seems that Axiom 11 should be mentioned then.

Part II: Write something reflective about the reading.

I really liked the example on page 43 where they show a set T={r,s,t,z} and that they define it in a way that it is considered a ring. It is so crazy but yet it makes sense because it follows the definition. Since Axiom 9 and Axiom 10 were mention right after the definition of a ring that there aren't many examples of just rings without Axiom 9 and 10, but this example at least showed a ring that isn't even a commutative ring, and I noticed that before I read that part of the reading so I was making sense of everything pretty well.

2.2, due on January 11

Part I: What was the most difficult part of the material for you?

     The most difficult part was when they used an example to show that addition and multiplication works with those congruence classes in the integers modulo n. This was on the first page of the section and I wanted them to explain it more. I feel like there is a lot of ambiguity. If they broke it down from congruence  classes, then to integers, and back into the classes, I would believe the example.

Part II: Write something reflective about the reading.

     I think it is really cool that many of the properties that work in the integers carry over in the integers modulo n. I feel like a whole new world has been opened up and there is so many new things to discover about the integers modulo n. It will seem difficult though to follow the changes the  books make in representing the congruence classes and to remind myself that they are classes, not integers. 

2.1, due on January 9

Part I : What was the most difficult part of the material for you?

       The most difficult part of the material for me is the connection between Corollary 2.5 to Theorem 2.3 and to the definition of congruence class. I can connect the definition of congruent to modulo and the remainder idea, but I am having a lot of difficulty connecting everything together. I think what makes it so difficult is to use so many variables rather than actual numbers to see the pattern. The congruence class is an abstract idea for me right now because I haven't written it out much yet, but hopefully as I work through the homework and look through our notes in the future I will get use to it.

Part II: Write something reflective about the reading.

      This section reminds me of learning about modulo in Math 290. That is probably why I can connect some of the pieces together but since it has been a little while, I am getting a little confused and lost. In Math 290 I learned more about the connection of how modulo relates to the remainder in the equation a=qn+r where r is the remainder. Yet I didn't go in depth about modulo arithmetic in 290 so hopefully I will get better with it in this class.

1.1-1.3, due on January 6

Part I : What was the most difficult part of the material for you?


     The end of 1.2 was the most difficult part of the material for me. I was definitely getting lost in all the equations and the implications and theories, but at the end of 1.2 I was confused about Theroem 1.6 The Euclidean Algorithm. I am still a bit confused on it. I understood it well in class but reading the actual statement of the theorem doesn't make sense to me with how the book writes it out. I am mostly confused with the second sentence which states, "If b|a, then (a,b)=b." I am confused because I can think of an example such as 3|27 but (3,27)=9 not 3. This also satisfies the first statement which is that a and b are positive integers with a>b or a=b, and in my example 27 > 3. So I just don't understand that statement.


Part II: Write something reflective about the reading.

     I think the most interesting part is the proof for Theorem 1.11 The Fundamental Theorem of Arithmetic. I liked it because it looks long and complicated but it makes complete sense. I also feel like it will lead into a lot of important proofs in the future so it seems like this is an important theorem and proof to remember and memorize. This whole section of reading goes into the idea of number theory which really fascinates me, but I guess the ending just had a good punch to it.  

Introduction, due on January 6

• What is your year in school and major? I am a junior at BYU majoring in Math Education.
• Which post-calculus math courses have you taken? I have taken Math 290, Math 313, Math 314, Math 334, Math 341, and Math 362
• Why are you taking this class? For the most part, I am taking this class because it is required for my major. I don't really know what abstract algebra is about but I do hope to gain a deeper understanding of the mathematics that will be most helpful for me in this subject to help me teach junior high and high school students better. I believe that this class involves a lot of proofs and I hope that I can make sense of the proofs and improve on writing proofs.
• Tell me about the math professor or teacher you have had who was the most and/or least effective.  What did s/he do that worked so well/poorly? One professor I had that was most effective was one that answered questions that the students had and that was patient whenever I would ask for help in his office hours. I always do better when I can visit with the professor and discuss the math with him. I also think that the simpler and the clearer a subject is taught in, the better I understand the concept.
• Write something interesting or unique about yourself. I really like the show Buffy the Vampire Slayer and I like to do puzzles. My best days are when I do them both at the same time.
• If you are unable to come to my scheduled office hours or the TA's scheduled office hours, what times would work for you? I am not able to come to your office hours, but I can make it to some of your TA's office hours time. I would prefer to still talk to you at some point and the best times for me are 12:00-12:50 MWF and 3:00-3:50 MWF or any time on T, Th.