Section 9.4

Difficult: It was hard for me to understand Theorem 9.31. I don't get what this theorem is saying or how it will be important. The proof doesn't really make much sense to me either, and I don't understand how it proves the theorem.

Reflective: I think it's interesting that so much can be done with integral domains, and I think the concept of a field of quotients is really interesting. It makes sense to think of the rationals as a field of quotients of the integers.

Section 8.4-8.5

Difficult: The most difficult thing for me to understand in these sections was the proofs of the Sylow Theorems, especially the proof of the second Sylow theorem. I don't understand some of the steps they take in the proofs.

Reflective: I think the idea of conjugacy is really interesting, especially conjugacy classes. It's interesting how they fit into the classification of groups problem, and the proofs of the Sylow Theorems.

Section 8.3

Difficult: I am having a hard time understanding the second example on page 264. I am having a hard time following their steps to get to the point where they applied Corollary 8.16.

Reflective: I think it's really interesting that there are so many properties that can be found from finite groups. This section was really interesting to me because I think that the Sylow theorems are interesting, and I look forward to seeing how we will use them.

Section 8.2

Difficult: This was a hard section! I think I kind of understand most of it, but I feel pretty shaky on what exactly a the group G(p) is and what a  p-group is. I also don't really understand the proof of Lemma 8.6. Also, I wish I knew why they kept switching between additive notation and multiplicative notation.

Reflective: I think this section is really interesting because it determines how we can classify all abelian groups. I also think it's interesting that most abelian groups are additive.

Section 8.1

Difficult: I am having a bit of a hard time understanding what exactly Theorem 8.1 means. I understand the example before it, and I think that the theorem is just generalizing this example, but I don't understand how we will be using it.

Reflective: I think it's interesting that some finite groups can be classified as isomorphic to groups of ordered pairs, or direct products. I also think that it's interesting that if M and N are disjoint subgroups in G except for e, then the elements of M commute with the elements of N.

Section 7.10

Difficult: This was a hard section for me to understand. Specifically, I don't really understand why alternating groups are so significant and why it makes such a difference that a subgroup is classified as alternating.

Reflective: I think it's interesting that there are so many properties that can be found from simple groups and alternating groups, and I think it's interesting that every element of and alternating group is the product of 3 cycles.

Section 7.9

Difficult: I am having a difficult time understanding how they got all the different factorizations for permutations on page 233. I also don't really understand the proof of Lemma 7.49.

Reflective: I think it's interesting that this new cycle notation for permutations can show us so many more properties of permutations that we couldn't see before.

Section 7.8

Difficult: This was a hard section. I understood the first half pretty well, but once I got to the Third Isomorphism Theorem I didn't really understand much after that. It seems to me that while we can prove all these theorems, they seem so ambiguous that they will have very little use for me. I am curious to see how we will use them.

Reflective: It's interesting that the First Isomorphism Theorem and the other two theorems before it can be applied so easily from rings to groups. I am interested to find out how we will use all the theorems that we learned in this section.

Section 7.7

Difficult: I am having a hard time understanding Theorem 7.38 and what uses it will have. It seems that if we wanted to use this theorem to find out if G is abelian, it would be easier to just show directly that G is an abelian group.

Reflective: I think it's interesting that the structure of a group can be discovered if the structure N and G are known. I also think it's interesting that there are so many properties that seem to be parallel to the properties of quotient rings.

Section 7.6 Part 2

Difficult: I don't understand what the difference between parts (2)/(3) and (4)/(5) are in Theorem 7.34. I know that the book explained a bit after the theorem was given, but I don't understand what it means and why these statements are not saying the same thing.

Reflective: I think it's interesting that a subgroup has to be normal in order for Theorem 7.33 to work. I think that the concept of normal subgroups is interesting because it seems like it would be much more unlikely that many normal subgroups, except for the center, would exist in a large group.

Section 7.6 Part 1

Difficult: I am having a hard time understanding the notation for congruence classes, both left and right. I don't really understand what the different sets on page 210 is under that definition for a left congruence class mean.

Reflective: I think it's interesting that there are some subgroups, called normal subgroups, that commute with every a in G.

Focus on Math Talk March 13

Difficult: I didn't really find anything difficult about this presentation, because it wasn't really about any concepts in math, just about the history of numbers as an idea. It was a really interesting to think about how numbers and counting came about. The information was very well presented, and everything was explained well so I was able to understand it all.

Reflective: I thought it was really interesting that so many words that we use today that don't even necessarily have much to do with math now came from mathematics, such as the word "stockholder." It was interesting to see how the first form of written language was to record mathematics, and I loved how he related this to everything else that we learn about every day, such as language and business.

Section 7.5 Part 2

Difficult: I am having a little difficult time understanding how they got the operation table in the proof for Theorem 7.30. I don't really understand some of the claims that the author makes when showing this part of the proof.

Reflective: I think it's really interesting that groups with prime number orders are cyclic and isomorphic to Z mod p. I also think it's interesting that this doesn't seem to apply only to prime numbers, but also to groups of order up to 8.

Section 7.5

Difficult: I am having a difficult time understanding Theorem 7.25. I don't understand why G is the union of all cosets of K, and how there can be a bijective function from K to Ka. It seems like Ka would be a lot smaller than K, so how can there be a bijection?

Reflective: I think it's really interesting that subgroups of a group G can also have congruence classes. I think it seems more similar to congruence modulo an ideal as opposed to congruence modulo an integer, so I hope that I will be able to understand it better this time.

Section 7.4

Difficult: I am having a difficult time understanding why the image of a function is important, what it will be used for, and what exactly part 4 of theorem 7.19 means. Also, I don't understand Cayley's Theorem and the proof of it.

Reflective: I think that the idea of 2 groups with different operations being isomorphic to each other is really interesting. It seems that isomorphisms of groups are more versatile than isomorphisms in rings, and I think they will be interesting and perhaps a little easier to work with, now that I understand isomorphisms better.

Test 2 Review

I think that the most important topics we have studied have been rings and groups. Within rings, we focused on congruence modulo n, congruence classes, polynomial arithmetic, and ideals. These are the main topics that we have covered, and they are the most important. Theorems that are important are the theorems that involve ring properties, subrings, and fields.
On the exam, I expect to see questions that test my knowledge on rings and fields, basic group properties that we have learned, proving that a set is a ring, group, field, or integral domain, isomorphisms, giving examples of different rings and groups, and proving that a subset is actually a subring or a subgroup.
For the exam, I need to study more examples of all different types of rings, fields, and groups. I also need to make sure I know the definitions in the book and that were emphasized in class. Some concepts I have trouble with are how to prove that two rings or groups are isomorphic, what a cyclic group is, and ideals and cyclic groups generated by a finite number of elements.

Section 7.3

Difficult: I am having a hard time understanding how the generators of groups work, and what Theorem 7.17 means. Specifically, I don't understand what exactly the set <S> is and what it means by "all possible products, in every order."

Reflective: I think that cyclic groups and the center of a group are interesting concepts, and I wonder what the connection between the two types of groups are, if there is any. I also wonder what the center of a group is used for.

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.

Section 4.2

Difficult: The thing that I think will be most difficult will be actually doing the problems. The one in the example in the book seems quite messy, and I am worried about doing the problems myself because it will be easy to make mistakes and get completely wrong answers. Other than that, the reading material was relatively simple because the math basically the same as in the integers.

Reflective: This material is almost exactly the same as in Z, and I am excited to learn how to do problems and find the gcd of two polynomials. It's helpful that the Euclidean Algorithm can be applied to F(x).

January 27 Questions

The homework assignments for this class take me anywhere between 1 and 2 hours. The reading usually helps, and the lecture are really helpful for clearing things up that I didn't understand in the reading. Doing the homework has definitely contributed the most to my learning, the exercises in the book are really helpful for me because that is how I am able to internalize the concepts that I learn from the reading and lecture. I usually find that I don't fully understand the reading until I have done the problems, and I usually also feel confident that if there is something I don't understand about the reading, then I will be able to understand it better when I do the homework.

Section 4.1

Difficult: I found it hard to follow the proof of Theorem 4.4, the Division Algorithm in F(x). It was helpful to have the examples, but I think that if I was able to follow the proof as it was being done I would be able to follow it and understand it better.

Reflective: I thought it was really interesting to learn about how the Division Algorithm can be applied to functions. Also, I think it's cool how there is a rule that can be applied to addition and multiplication of functions. I am looking forward to working with functions while doing problems, although I think it might be a little messy.

Section 3.3

Difficult: It was slightly difficult for me to understand what the book meant when it said that a property is preserved by isomorphism. The examples clarified it a little bit for me, but I still have trouble with an arbitrary property that can be preserved through an isomorphism, and which properties those are.

Reflective: It was interesting to learn about isomorphisms. I learned about them in linear algebra, although I don't remember the third property that this book identifies as being one of the properties of an isomorphism that we learned about in linear algebra. Additionally, I remember learning about bijections in math 290, so it's nice to have some background about this subject to rely on.

Section 3.2

Difficult: I found it a little difficult to understand why subtraction can be used to prove that a subset of a ring is a subring. It's really interesting that part one of Theorem 3.6 accounts for 3 parts of Theorem 3.2, and this was difficult for me to understand at first because I had trouble seeing the connection between the 3 parts of Thm 3.2 and subtraction.

Reflective: I think it's interesting that rings have so many properties in common with the integers. I am looking forward to learning more about fields because I think that they have a lot of interesting properties, and seeing how those properties relate to rings in general.

Section 3.1 Part 2

Difficult: I found it hard to understand the example and theorem about Cartesian products. I've never really been able to understood what they were and how they work.

Reflective: Once again, I think that the idea of a subring is similar to the idea of subspaces of vector spaces in linear algebra. I think it's really interesting that the subset inherits some of the axioms from the ring, such as commutative addition.

Section 3.1

Difficult: The concepts of rings and fields are slightly difficult to understand at first because I'm not sure how they will be used and applied. I want to know what their relationship is to the integers. I also don't know what they mean by "abstracting" algebra.

Reflective: I think that these rings are somewhat similar to vector spaces that we learned about in linear algebra. They are both sets of arbitrary elements that have a certain number of axioms to satisfy. I hope this will come in handy in the future as I am trying to understand what rings and fields are.

Section 2.3

Difficult: I understood all the proofs in this section, but I am having a hard time applying them to examples and seeing how they will come in useful. For example, I understand how the proof of Corollary 2.9 works, but I wish there was an example in the book that would tell me how to use it.

Reflective: I think it's so interesting that the prime numbers create a special case and have special properties. Also, the fact that a and n being relatively prime makes a difference is also really cool. I've never known that prime numbers have so many special properties, and I am excited to learn more about them.

Section 2.2

Difficult: It was a little difficult at first for me to understand how exactly addition and multiplication worked in congruence classes. I was confused by the use of 4 integers a, b, c, and d in Theorem 2.6 and its proof, and then only 2 integers in subsequent examples. However, I think I was probably just overthinking it and I understand how they both work now.

Reflective: I think it is really interesting that Zn is so closely related to Z. Also, it's interesting that there is a way to add and multiply congruence classes, and I hope that we will be able to do more work with these in the future so that we can find out how to apply congruence class addition and multiplication.

Section 2.1

Difficult: It was a little hard at first to wrap my mind around the concept of congruence, even though I've studied them before. I have to think hard to be able to understand what the examples mean and what congruence modulo n means, but I think this will get easier after practice.

Reflective: I remember learning about congruence in Math 290. I think it's really interesting that congruence modulo n has similar properties to the relation of equality. Also, I think Corollary 2.5 Part 1 is interesting, that [a] = [r], with r being the remainder when a is divided by n.

Sections 1.1-1.3

Difficult: The most difficult part of the material was understand the proofs given in the text, particularly the proof of Theorem 1.1 in Section 1.1. There were parts of the proofs that I had a hard time understanding because I couldn't see the connection between two steps. This has always been difficult for me.

Reflective: I think that the Euclidean Algorithm is very interesting. Last semester I took a math education class and we learned about how to teach the greatest common divisor, and I love this way of doing it. I think it's so interesting that the numbers are all so closely connected, and the the common factors of two numbers can be expressed as a linear combination of those two numbers.

Introduction

I am a sophomore and my major is mathematics education.
I have taken Math 290, Fundamentals of Mathematics, and Math 313, Linear Algebra
I am taking this class because I have to for my major and because I want to become better at proving mathematical concepts and theories. I am not very good at proofs right now, so I hope this course will help me.
I have had many outstanding math teachers, but my 290 professor here at BYU stands out. I wasn't very good at the work we had to do for the class, but his door was almost always open and I could come ask him about the homework. This was very helpful since there wasn't a TA for the class, and I was often in his office asking him for help.
Something unique about me is that I am studying Chinese and I have a little brother who is adopted from China.
Your office hours work well for me, as do the TA's office hours.