Section 7.2

Difficult: I am having a hard time understanding what exactly the order of an element of a group is. I know what the order of a set is, but I don't understand how the two different orders are related, or if they are at all.

Reflective: It's interesting to me that, while there are some group properties that are almost the same as the ring properties, there are also many properties that are very different and have no relation to rings. for example, the properties of the order of a group G don't have any relation to rings.

Section 7.1 Part 2

Difficult: This section was relatively simple to understand because it is very similar to rings and fields and most of the reading was examples. I think it will be slightly confusing to work with dihedral groups of higher degree, though.

Reflective: I think it is interesting that rings can only be groups under addition, and not fully under multiplication. I am interested to look at more examples of groups because they are more inclusive and I think they will have some interesting properties.

Section 7.1 Part 1

Difficult: I am having a bit of a hard time understanding why there is such a big difference between groups and fields. I understand the examples in the book, but it seems to me that all rings can also be considered groups.

Reflective: The concepts in this section seem almost the same as with rings, except that there is only one operation defined in a group. Groups seem more like fields, however, because every element of a group G has to have an inverse.

Section 6.3

Difficult: I am having a bit of trouble understanding why, if P is prime, then R/P is only an integral domain and not a field.

Reflective: I think the idea of prime ideals is really interesting. I am also interested to find out more about maximal ideals and why they can form fields with R/M but why R/P cannot.

Section 6.2

Difficult: I am having trouble understanding Theorem 6.13, the First Isomorphic Theorem. I am probably still misunderstanding quotient rings, so I don't get why R/K is surjective to S if f: R --> S is a surjective homomorphism.

Reflective: In linear algebra, we briefly discussed kernels, and they make more sense this time, although I am still somewhat unclear about this whole section. I think they are an interesting concept though, and it will be interesting to learn more about what properties they have and how they can be used.

Section 6.1-6.2

Difficult: I don't really understand the idea of congruence modulo an ideal. I also don't understand what the difference between a coset and a regular congruence class is. I think they are about the same thing but I'm not sure, I think I just need to understand what congruence mod an ideal means and then I will understand.

Reflective: I think the idea of a quotient ring is interesting. It's cool that there are so many different types of rings and there are so many different sets of numbers that form rings.

Section 6.1

Difficult: I am having a hard time understanding Theorem 6.3, and why this type of set is an ideal. It seems to me like you could multiply the largest number (because the set is finite) by any element of R and have an element that is not in the set, so therefore it is not an ideal, so I don't think I understand exactly what the theorem is saying.

Reflective: I think it's interesting that there is another set of numbers that have even more properties than rings and fields. I am interested to know what other properties ideals have and how we will use them.

Section 5.3

Difficult: I don't understand extension fields very well. I understand what they are, but I don't understand why a polynomial that doesn't have a root in F[x]/p(x) can have a root in the extension field K. I also don't understand what the use of this extension field will be.

Reflective: While I don't understand extension fields, I think it will be interesting to learn about them and understand how they work and how they will be used. I also am interested by the fact that an extension field K is isomorphic to the complex numbers C.

Section 5.2

Difficult: It is difficult for me to understand why exactly there is a subring of F[x]/p(x) that is isomorphic to F. I believe I understand the proofs in the chapter for the most part, but this concept is still giving me trouble.

Reflective: I am glad to know that addition and multiplication in the congruences classes in F[x]/p(x) are the same as regular addition and multiplication with polynomials. Also, I think it's interesting that the congruence classes in F[x]/p(x) are even stronger than in Z mod n.

Section 5.1

Difficult: I am having a hard time relating the integers and F[x] in terms of congruence and congruence classes. The examples at the end of the section were really helpful, but I still feel unsure about why some fields don't have infinite congruence classes, and what determines the number of classes they will have.

Reflexive: I think that once I get the hang of working with congruence classes in F[x], I will be able to understand better the connection between them and the integers. I think it's interesting that there are so many connections between the two rings.

Sections 4.5-4.6

Difficult: It was hard for me to understand some of the different ways to find roots and check irreducibility in Q[x]. I also don't understand if, in some of the theorems, some of the conditions apply and are proven true, if f(x) can still be irreducible.

Reflective: This was interesting material because it talked about how to find roots, and this is something that I can apply to my other math studies really well. I though it was interesting to learn about how to find roots in the real numbers and complex numbers, and I am looking forward to doing problems to find roots using these different theorems.

Section 4.4

Difficult: I think it will be hard to understand the difference between when f(x) is a polynomial and a polynomial function. I understand what the difference between the two is, but I'm not sure how to apply either of them or how they will be used, and I don't know how to tell the difference between them.

Reflective: This reminds me of high school math classes a bit, when we had to find the roots of functions. I think it will be interesting to work with functions and find their roots in this new light of fields and Z mod n.

Section 4.3

Difficult: I had a difficult time understanding most of this section, however probably the most difficult was the concept of associates. I think this might be something rather simple and I am probably just overthinking it, but I don't really understand exactly what they are and how they are used in the subsequent proofs in the section.

Reflective: I think it's really interesting that all nonconstant polynomials can be written as the product of irreducibles. It had never occurred to me that this might be the case. It is so cool that there is so much overlap between polynomials and the integers, and yet still some key differences as well, such as the units in the polynomials.