Section 7.2, Due March 1

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

Theorem 7.8 was a lot to take in, since it really is like 4 theorems. When reading the proof for 3 and 4, I was really confused and I think it will help to actually go over them in class. The proof for part 2 was a little confusing as well, but it makes sense why proving 2 would also prove part 1. 

Part II: Write something reflective about the reading.  

The finite order of an element a in a group is a really interesting idea. I first tried to think of examples to make sense of what this means, but in the integers and rational and real numbers, there is no number that has finite order. Then, when I read the examples in the book, it made sense of what kinds of elements would have finite order.  

Rest of Section 7.1, Due February 29

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

I can understand the part of how a ring is an abelian group under addition, but I think I am still confused on how it doesn't work with multiplication. And since a ring is an abelian group under addition, does that mean that the binary operation is addition and you only need that operation. I guess I am still confused on what a binary operation still means. 

Part II: Write something reflective about the reading. 
 
When they book started taking about a cut out square and rotating it and reflecting it, I thought of the example in class with the Rubik's cube. It is really cool to make that kind of connection and to see how both are groups.

Section 7.1 first half, Due February 27

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

The most difficult part of the material is that a group has a binary operation *. At first I didn't think much of what that was saying, but then when they went into the examples of compositions of functions, I realized that the composition was the binary operation, which seems weird. 

Part II: Write something reflective about the reading. 

I am in a combinatorics class right now and we are talking about permutations, so I could easily see the connection of why the order of S(n) is n! and that the notation for these functions. It was really cool to make the connections of this section to my combinatorics class.  

Section 6.3, Due Februray 24

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

The section about maximal ideals really confused me. After thinking about it for a while, the example given with (3) kind of made sense how (3) is a maximal ideal, but then the example of Z[x]/(x) really lost me and I was just lost on what that even meant. 



Part II: Write something reflective about the reading.

It is becoming really difficult to tie everything from what I have learned in previous sections to this chapter. I am still struggling with the idea the Z/(3) is the same as Z mod 3 and I just have a really hard time trying to figure out the cosets of R/I for some example. It takes a lot of reflection to really understand this.   

Rest of Section 6.2, Due February 22

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

 I am most confused about the proof in theorem 6.11. The second part of the proof is assuming that f is injective where f:R to S and f is a homomorphism of rings with kernal K, and we are trying to prove that K = (0subR). I can understand most of it until they say that f(c)=0sub s and that f(c)=f(0subR). I do not know why f(c)=f(0subR). I kept trying to look at how the kernel fits into this equation and I am just getting really lost at this point.


Part II: Write something reflective about the reading.  


This section is really becoming more and more abstract. I am having such a hard time with making a connection with some of this stuff to things we have already done, but maybe there are no connections to really make yet from past things I have done. I just hope that working on the homework and seeing other example will help me understand what some of these theorems are even saying.

Section 6.1-6.2, Due February 21

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

I am most confused about the set R/I. I feel like I understand what an Ideal is and how it is related to R, but now connecting it to modulo of R seems too confusing and abstract. I don't think I am comfortable yet with cosets and that is probably where most of this confusion is coming from.


Part II: Write something reflective about the reading. 


The notation in this section starts to get really confusing. Since a+I denotes a congruence class yet they then define addition in a similar way and it is strange. I don't really understand why they didn't use a different notation for addition or for the congruence class, like something that related better to Chapter 2.

Section 6.1, Due February 17

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

I was most confused on the example given for what an ideal isn't. While the concept and term of ideal seems to be understandable, the book didn't seem to give a concrete simple example of a principal ideal generate by some number.


Part II: Write something reflective about the reading. 

While I have never heard of the math term ideal before, the book seemed really clear about what it was and it made a lot of sense with the example they use at the beginning of the chapter. Normally I don't understand new math terms very well at first, but this one was quite easy to grasp.

Section 5.3, Due February 15

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

 I am most confused about the definition of an extension field and a subfield. With how they used subfield, I felt like it should have been used the other way, like K is a subfield of F. And so that is why I am even more confused with the extension field definition. It doesn't make sense to me with how it has been for subrings.

Part II: Write something reflective about the reading.

  At first I thought it would be too obvious to say that F[x]/p(x) is a field if p(x) is irreducible because F[x] comes from a field, but then I realized I was generalizing too much. Even though F is field, I guess it is still possible that F[x]/p(x) is not a field when p(x) is reducible, which is really crazy.

Section 5.2, Due February 13

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


I was most confused about Theorem 5.7. The second sentence was way too confusing because it used a symbol for mod and then had the word modulo and I didn't understood how those related to each other, and I still don't. Maybe the book just misprint it incorrectly or I am just not understanding the notation correctly, but I don't understand what the theorem is saying in English. 


Part II: Write something reflective about the reading.


It wasn't until I started reading this section when I realized that F[x]/p(x) is similar to the idea of integers mod n. Once I understood that notation in that way, it started to click better. When I first saw it, just knew what you would say if you saw that notation, but I didn't really understand what I was saying. Polynomials are a lot harder than integers to understand for me. 

Section 5.1, Due February 10

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


I understand what congruence class is, but I am so use to it in the integers mod that I am a little confused about how it is in a polynomial ring. Even understanding mod in a polynomial ring just seems so much more complicated.



Part II: Write something reflective about the reading.

It is really cool to see how each section of previous chapters relates to a different structure of "numbers" and I can clearly see that in this section. Many of the theorems are worded very similarly and it is helpful to look back at the previous sections to make those connections. 

Section 4.5-4.6, Due February 8

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


I am lost at the start of theorem 4.20 (Rational root test) because it discusses about the variables r and s, but I don't even understand where they got an s! I know that is comes from the root of f(x) but I guess it is worded strangely and I would need an example that I work on to see the connection.



Part II: Write something reflective about the reading. 

Theorem 4.25 I think is really amazing because it seems impossible that EVERY non-constant polynomial in the C[x] has a root in C but after seeing the proof and thinking about it, it makes sense. But still, it is surprising.  Also,  many of the theorems after had similar ideas with finding things dealing with EVERY non-constant polynomials in a polynomial ring, which is really amazing.

Which topics and theorems do you think are the most important out of those we have studied?

The Euclidean Algorithm, the fact that we can write the gcd of f and g as a linear combination for some u and v, and the theorem that says "If F is a field, then for any f in that field polynomial and any a in F, the element a is a root of f iff (x-a) divides f." 

What kinds of questions do you expect to see on the exam?

I expect there to be proofs, definitions, and many some small calculations that we may have done similar in the homework. But I think the proofs will be proven with the theorems and definitions we have memorized. 

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I am still trying to understand better how to prove if a function is surjective or not. Most of the time it just seems obvious, so I don't know if I am missing something or if I understand it so well that I am making it more complicated.  One example is, problem 15 on 3.3 which says, Show that the complex conjugation function f:C to C (whose rule is f(a+bi)=a-bi) is a bijection.

Section 4.4, Due Feb. 6

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

 The most difficult part was differentiating between what x is as an indeterminate or a variable. I am just confused at what the phrase, "the function f is induced by a polynomial," which makes it hard to understand what an indeterminate is and what a variable is. I feel like I may be getting confused over something that is simple, so maybe it isn't as hard as I think it is.

Part II: Write something reflective about the reading. 

The Remainder Theorem is an interesting theorem that I am kind of shocked is a theorem. Yet when I look at the proof, it makes sense. I guess I need to try a few problems to believe it better for myself.

Section 4.3, Due February 3

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

The definition of an associate is when I started to get a little lost. I think it would have made more sense to me if the book had given examples that showed the idea. Then when I read the definition of irreducible, I was really lost because I had to understand what an associate is. The example afterwards helped me a little bit, but I am still most confused about what associate and irreducibility is.

Part II: Write something reflective about the reading.

I understand that his section connects to a previous section we did with the integers and prime, but the book didn't really focus so much on connecting the idea of irreducibility in with primes. I eventually saw some of the equivalences on my own which was interesting to make the connection myself. It would have been more helpful though I think if the book did connect these two ideas more so.