6.2 Part 2, Due February 28

Difficult: I don't really understand the first isomorphic theorem. I don't understand how to use and apply it to what we are working with. The examples on page 150 confused me even further. I think it will be difficult to find a function that is a surjective homomorphism, and to find the kernel, and use this information to find out the structure of quotient rings. I think mostly, I just don't know what this theorem even means! So knowing when it is applicable is going to be quite a challenge.

Connections: I like that ideals are related to homomorphisms as well. I like working with homomorphisms and isomorphisms mostly because I feel that the proofs with them are very computational... which I really like. I like that for most of these theorems, one of the givens is that it is a homomorphism. This makes going through the proofs fairly computational when we get to use this information. It is simple that homomorphisms are defined by two things, which are easy to show in a proof.

6.2 part 1, Due February 25

Difficult: I feel like the most difficult part of this section was still just understanding cosets. I felt like I understood pretty well how to add and multiply cosets, but I still don't get how you find the cosets of these rings. I feel like maybe the notation is a little foreign to me. I don't quite know how to find all of the cosets of the form a+I, so adding and multiplying will be quite difficult if I don't even know what the cosets are.

Connections: I thought that the multiplication and addition tables made sense to me. I feel like we have done a lot of addition and multiplication tables, so I think that once I know the cosets of the ring, I will be able to make the multiplication and addition tables as long as it is not an infinite set of cosets. However, as I mentioned above, I'm not convinced that I will be able to find the cosets... which is a slight problem if I need to try to add and multiply them together.

6.1 part 2, Due February 23

Difficult: For me, the hardest part of this section was finding the congruence classes. Just like with polynomials, finding congruence classes is very difficult. I have a hard time deciding how many congruence classes there are, and what they even are! I think however, that this section is more difficult than polynomials to determine the congruence classes. I don't quite understand the notation well enough to be able to find what the congruence classes are. Also, I don't really get when a (left) coset is either. How is this different than a congruence class, or is it the same?

Connections: I like that congruence is similar to the congruence that we have been dealing with in the polynomials, or in the integers. I like consistency in definitions, and so I like that this definition is very familiar. It is easier for me to remember when it follows the same pattern as the other kinds of congruence. I also like that theorems 6.4 and 6.5 are similar to the theorems in the other sections about how congruence is reflexive, symmetric, and transitive and that addition and multiplication can follow in congruence. These definitions are definitely something I have seem before, so its nice to have something familiar attached to the new ideas of ideals.

6.1 part 1, Due February 22

Difficult: I felt like in this section I understood the idea of ideals, until it started talking about the different kinds of ideals. I really liked theorem 6.1, and I thought it was pretty straight-forward. However, the principal ideals, and finitely generated ideals were a little confusing to me. I wasn't quite sure what the difference was between them, and how they differed from just being an ideal. I found it difficult to distinguish between them and understand why they were important.

Connections: As I mentioned before, I liked the idea of the ideals in the beginning. I felt like it sounded very familiar to stuff we have already learned this semester. Theorem 6.1 reminded me a lot of a subring! The properties to be an ideal seem to me to be a lot like the properties to be a subring. I like that the properties of being an ideal are that they are closed under subtraction and multiplication, even though it is a little different from a subring since in the multiplication, one of the elements need not be in the set I, but the product of an element from within the set and outside the set need only be in I.

5.3, Due February 18

Difficult: The thing I found most difficult in this section was the idea of an extension field. I don't really get the point of this. So it makes irreducible polynomials have roots in the extension field, but I couldn't figure out what those roots are, or why it matters. I was also didn't understand how the extension field can help us define complex numbers either. Basically, I just don't understand why extension fields are important, and what it helps us do.

Connections: So I feel like I am starting to get the hang of working with polynomials modulo p(x), but it is pretty different from modular arithmetic with integers. I also like that I am getting more practice with irreducible polynomials, and finding them. I like using the root test to find or verify that a polynomial is reducible or irreducible. This is a simpler way for me than trying to find all of the reducible ones and picking one that is not in that list. So it is nice to work more with irreducible polynomials in this section so that I can get more comfortable with them.

5.1, Due on February 14

Difficult:
For me, the most difficult part of this section was when it talked about the congruence classes. I think this would be so hard to find the congruence classes, and determine how many congruence classes there are. This section is going to be more difficult than congruence classes of integers because I think dividing polynomials is way harder! I am going to have a hard time finding congruence classes, and even knowing what things are of the same congruence class.

Connections:
I like that this section pretty much mirrors the section on congruence classes for integers. It follows the same kinds of properties and definitions as it did in the other section, but this time the dividing part will be much more difficult. It will be harder to recognize when something is in the same congruence classes because sometimes it is harder to see what is going on with polynomials. Hopefully my understanding of congruence classes for integers will help me understand it better for polynomials.

9.4, Due on February 11

Difficult:
I had a hard time just plain understanding what this section was talking about. I got lost at the relation ~, and I never really found my way back. I found it difficult to understand how they were deciding to define multiplication and addition, and I think this is because I was confused what they were doing with quotients and how they were turning them into an ordered pair and adding them. So much of this section built upon the beginning that I was confused pretty much throughout the whole section because I didn't even understand the beginning.

Connections:
I feel like since I know what an integral domain and a field are, I should be able to better understand this section by connecting the new concepts involving the quotients and applying them to integral domains and fields. I feel pretty comfortable with the definitions of these, so I think that I can use this knowledge to help me make sense of the new application using quotients.

Exam Questions, Due February 9

I feel like the most important thing that we have covered so far this semester is rings. There are so many different kinds, and I feel like a lot of the stuff we have learned since learning about rings involves rings, or integral domains, or fields. So knowing the definitions and how to use these things is very fundamental. Maybe that is why I think they are so important; because I don't really understand them that well, or how and when to apply them. Something that I think is important and that I will need to study a lot is the theorems about rings, fields, and integral domains. I need to know these theorems really well so that I will know when they will be useful to me on a problem.

I expect that on the exam, there will be questions involving definitions, maybe fill in the blank or something. I also think that there will be a lot of questions asking for examples of different things we have talked about in class. I think there will also be many proofs. The proofs are what make me the most nervous. I have a hard time understanding where to begin with most proofs, but once I can understand what the question is asking, I can usually figure out how to go about proving it. I also expect that there will probably be lots of questions that are similar to homework questions.

Basically, I think I need to work on understanding the past couple of chapters. I did fine with modular arithmetic and integers, but rings, and isomorphism stuff got really confusing for me. I have had a really hard time understanding what it is talking about and how to apply that to a proof or problem. I need to work on homework from these chapters and learning the theorems to be able to use them on the exam.

4.4, Due on February 7

Difficult
The hardest thing for me in this section was just the idea of the functions that are induced, creating polynomial functions. I can't really tell the difference between polynomials and polynomial functions. Also,  how to know if the polynomials are the same function even if they induce the different polynomials. How do you test for this? What number do you have to put into the function to decide if two functions are the same? I am still getting confused by the different meanings of x as well and how to know what x represents in the function or polynomial.

Connections:
This section connects the idea of irreducibility and roots, or factoring. I have been factoring polynomials for a long time, so this part is easier for me. But deciding when a polynomial is irreducible is still difficult for me especially when dealing with the real numbers or modular arithmetic. But, I am fairly comfortable factoring polynomials and finding roots in the integers, so now I just need to think about doing this in different classes of numbers.

4.3, Due on February 4

Difficult:
I had a really difficult time with just the definitions in this section. I don't really understand what an associate is. It sounds to me like it is just a multiple of another thing, but then it seems like all things would be an associate. I just feel like I need to see some examples to really comprehend what this is. The same goes for irreducible. I just can't quite understand what that means. I get that it is like prime, but I feel like if I saw some more examples and had some more exposure to it, then I would understand its meaning better.

Connections:
I found it hard to make connections with the information in this section. I found that theorem 4.11 sounded quite familiar. It seems a lot like some of the theorems for prime numbers. This makes me realize that irreducible is a lot like prime numbers, but I still get confused when applying this concept to polynomials. I guess irreducible polynomials are "prime" polynomials, but that is still confusing for me to relate polynomials to prime numbers.

4.2, Due on February 2

Difficult:
I think the hardest thing for me to remember is that the greatest common divisor is monic, or the leading coefficient is 1. I was confused about how you can just multiply by the multiplicative inverse of the leading coefficient to get it to be monic. I wouldn't think of doing that and I feel like that means anything can be a greatest common divisor by multiplying by the inverse of the leading coefficient.

Connections:
I really liked this section. I really like factoring, and I feel like that is what division of polynomials is. I have been factoring polynomials since I can remember in algebra, so it will be good to be able to see how that will apply in abstract algebra as well. I also like finding the GCD, so I am excited to try to find the GCD of polynomials by factoring and finding the highest common term.