Final Exam Questions, due on Wednesday, April 13

1. Which topics and theorems do you think are important out of those we have studied?
I think the following items are important: the Cauchy theorem, cyclic groups, normal subgroups, quotient groups, other types of groups: symmetric, alternating, simple, abelian, finite abelian, etc.

2. 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 in class on Wednesday.
I am still confused about things that deal with quotient groups, cyclic groups, and direct products. These are the three things that have not sunk in yet. I still do not know how to create cyclic groups, and I do not know how to apply quotient groups, and I do not know how to figure out direct products. Any problem that deals with one of these three questions would be nice to see on Wednesday. Like an example of writing a direct sum/product of certain groups would be nice to see on Wednesday.

3. How do you think the things you learned in this course might be useful to you in the future?
I do not think I am going to take any of the theorems or properties we have learned in this class in my future for teaching mathematics to middle school students. However, I do think I can take a more non-literal approach with what I have learned in this class and apply it in my future. I have learned how to work really hard, and I have developed my ability to analyze and think about problems and create my own proof. This class has taught me to really think about mathematics and construct mathematical ideas on my own. And I will be able to use these things that I have learned in my future with harder classes that might come my way and for when I am trying to get my future math students to construct their own proofs and properties for mathematics.

8.3, due on Monday, April 11

1. The hardest part of this reading to understand is theorem 8.15, which is the second sylow theorem. I do not know how if P and K are both Sylow's how P = (x^-1)Kx for some x in G. I do know see how that would work and it does not make much sense. Can't P and K just be differnt prime number-subgroups, so then they would not be related in that way? I'm confused.

2. The most interesting part of this reading was that there was such a classification as a Sylow p-subgroup. I like when new properties and terminologies are introduced to us. It makes me feel like people have really studied abstract algebra before us and have created the most important things you can do with groups so that I know these things are important and are not just random theorems and properties to random groups of numbers. So I like this and I like new things like Sylow p-subgroups and that is what I find so interesting about this section.

8.2, due on Friday, April 8

1. Although there were lots of proofs and lemmas in the section to go through and understand, the hardest one to grasp was theorem 8.7 - the fundamental theorem of finite abelian groups. It makes me think that this theorem is most important because its name is the "fundamental theorem." However, I think it is really confusing that it is so important and fundamental to know about finite abelian groups that they are direct sums of cyclic groups, each of prime power order. Why is this so important? And why do we care? I do not understand.

2. The most interesting part of this section was that there was sooo many different proofs and properties for finite abelian groups. I would have never thought they were so important or unique in the sense that they have all these special theorems that people have discovered, studied, and now teach to anyone in abstract algebra. It is pretty cool.

8.1, due on Wednesday, April 6

1. The hardest part of this reading is understanding some of what theorem 8.1 is saying. I do not understanding G's elements being of the form a1a2a3... with each one pairing up with N1, N2, etc. What is an example of these types of elements in G and their corresponding normal subgroups in actual numbers and groups and not just letters. I cannot follow anything with letters if I do not have an example of what it is saying in numbers. It is just very confusing.

2. The most interesting part of this reading was that you can write groups as a sum of two or more of its subgroups. I think that is really cool and interesting. Also, you can know how many elements are in a group made of up crossing between other group because if those other groups are finite, you just multiple each's order to get the order of the cross of them all. That is really cool too. I like doing fun stuff like that.

7.10, due on Monday, April 4

1. The hardest part of this section to understand is the three cases in the main proof of theorem 7.52. Once we start having elements that are products of cycles written as (123...) and lots of them together, I start getting really confused and I cannot follow the work of the proof. So this is the hardest part to understand.

2. It is interesting that this section is basically just one theorem and its proof (with a couple of lemmas and their proofs included). This tells me that this theorem is so important that an entire section and day in class needs to be dedicated to it. So, I am determined to understand this theorem well enough to appreciate its importance in our study of abstract algebra, specifically in our understanding of the classification of finite groups.

7.8, due on Wednesday, March 30

1. The hardest part of this reading is understanding the importance in some of the theorems discussed and proved in this section. Some of them may make sense, but I wonder why they are important and why we have to learn them. Like, for example, why is it important to know theorem 7.45: G is a simple abelian group iff G is isomorphic to the additive group Zp for some prime p. Also, why is it important to know that if you have a situation like theorem 7.44 part 3. It just seems like these things are not very pertinent to our study of mathematics, thus it becomes very difficult for me to learn these things and understand them when I think they are somewhat not very useful.

2. The best part of this reading is being about to create homomorphisms and isomorphisms with/to quotient groups. I love coming back to concepts I actually understand and I actually understand homomorphisms and isomorphisms! This makes the new math for the day better because I feel like I have a strong handle on some of the new stuff because it is old stuff that I really understand still.

7.7, due on Monday, March 28

1. The hardest part of this reading is keeping straight all the different applications to quotient groups such as cyclic, abelian, normal, orders, and cosets. It is a lot of different properties of groups to be pushed into just one little section. Quotient rings are hard enough, thus making quotient groups hard, thus adding all the properties of groups to quotient rings is very confusing and hard to commit to memory and understanding.

2. It was really interesting that quotients came back into the class. We had studied quotient rings but now we are studying quotient groups. I like that this connection exists and that quotient things are coming back. Although, I was never very good with understanding quotient rings, so I probably will have to work hard to understand quotient groups.

7.6 Part 2, due on Friday, March 25

1. The hardest part of this reading for me is why in Theorem 7.34 - the 5 equivalent statements about normal subgroups - are statements #2 and #4 necessary as well as #3 and #5. Why do we need subsets and equal to for aNa-1, as well as with a-1Na. Wouldn't it be sufficient to just have statements 1-3 because 4 and 5 are included in 2 and 3? This is why I am confused and why this is the hardest part of the reading, because I do not see a purpose in this.

2. The most interesting part of this reading is that just because a subgroup N of group G is normal so Na = aN for every a in G does not mean that na=an for every n in N. This is fascinating to me. It is almost like a subgroup is normal is everything in G gets hit from both the left cosets and the right cosets but not necessarily that the things hitting are the same cosets from left and right. This is a really interesting idea to think about and this is what I liked learning the most in this section.

Extra Credit - Tuesday, March 22

Deanna Haunsperger - Carlton College - "Bright Lights on the Horizon"

1. The hardest part of this lecture was understanding exactly what the theme/purpose/topic of her lecture was. Every few minutes she started a new snippet about some interesting mathematics that someone is doing. But I did not know how each related to each other and what the point of talking about each of these people was other than it was really fascinating. I think she was just mentioning interesting articles that have been published in the journal she is in charge of, but it was still hard to follow and confusing to switch between all these different stories every two minutes.

2. The most interesting part of this lecture was EVERYTHING! She was so interesting to talk to and I actually understood what she was saying! The parts of the lecture I enjoyed the most were the parts where she would talk about mathematicians who did work that dealt with the arts. For example, mathematician Stephen D. Abbott studies "Turning Theorems into Plays." This is so cool because I am in love with plays and I love plays with math. She mentioned that the main playwright Abbott studies is Tom Stoppard. I actually saw one of his plays in London last summer, but I am ashamed to say that I did not know that Stoppard wrote other plays about math. She mentioned the play "Arcadia." I am now inspired to go read it! I also loved her talking about the topic "The Eccentricies of Actors." This deals with the Bacon Number and if Kevin Bacon is the center of the unverise and how many connections it takes for an actor to be connected to Kevin Bacon. Then she related it to Erdos numbers in mathematics. I absolutely loved this lecture. I learned things, and I was inspired by things she said, and it was just so interesting!!

7.6 Part 1, due on Wednesday, March 23

1. The most difficult part of this reading is keeping track of the left and right cosets. I already struggle with finding cosets, so now having to understand right and left cosets and be able to find them and compare them to see if they are different so that you can figure out if a subgroup is normal or do the right and left congruent principle is really confusing. I am worried that this is just going to make cosets even harder for me to understand.

2. The most interesting part of this section was that subgroups can have elements that are left congruent or right congruent mod the subgroup. This is so interesting. I have never thought about a subgroup or anything we have discussed having two different congruences - a right one and a left one. This is a nice layer added onto to the information we have already learned.

Midterm #2 Questions, Monday, March 21

1. Which topics and theorems do you think are the most important out of those we have studied?
Groups, subgroups, ideals, rings, and subrings. Knowing the properties and examples of these are going to be the most important.

2. What kinds of questions do you expect to see on the exam?
I expect to see questions about examples of different groups and ideals and such that we have talked about. I also expect to see a couple of proofs, maybe one or two we have done in class, and then a couple that we haven't done exactly but are similar -- like proving if something is a group, subgroup, finding ideals, kernels, etc.

3. 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 in class on Monday.
I would like to review ideals. I have forgotten a lot of chapter 6, and that would be the best thing because other earlier principles build on newer stuff, but ideals would be something good to go over because I am still confused. I would like to see a problem of listing off all the big examples of groups, subgroups, etc.

7.5 Part 2, due on Friday, March 18

1. The hardest part of this reading was understanding the new theorems about how every group of 4 is isomorphic to either Z4 or Z2 x Z2 and the theorem about every group of order 6. I guess I do not understand why these theorems are important, and the proofs do not convince me that it is true for every group of order 4 and order 6.

2. The most interesting part of this reading was bringing back the principles about primes for theorem 7.28. It was cool to apply primes to groups.

7.5 Part 1, due on Wednesday, March 16

1. A lot of this reading was not difficult because it was just applying congruence stuff and cosets to groups now. So the hardest part was just the new terminology and symbols for the index of H in G is denoted [G:H]. This is kind of confusing for me just because it is something that is actually new in this section. So then using that and putting it into Lagrange's theorem is kind of confusing because I am having a hard time wrapping my mind around what it means and how to use it properly.

2. The most interesting part of this reading is that that theorem 7.26 is named Lagrange's theorem. I am curious to know how this relates to like Lagrange multipliers that I have studied before (in, I think, linear algebra but I'm not sure). Is it just the same guy who discovered both of these? Or are they connected principles to each other? I don't know because I don't actually remember Lagrange multipliers other than the name is something I have studied before. So I would be interested in learning the connection between that and the Lagrange's theorem in the reading.

Extra Credit - Tuesday, March 15 - Vitaly Bergelson's Lecture

1. The most difficult part of this special guest lecture to understand was where exactly everything he was saying fits into mathematics, particularly to my learning of mathematics. While many of the terms and steps he did in the proofs he was showing were familiar to me, I did not understand what he was trying to do. Additionally, I have a really hard time understanding many people that lecture, including him. So when I can't understand what they are saying, I have a hard time following their thoughts and knowledge.

2. The most interesting part of this lecture was the fact that it was the first mathematics guest lecture I have attended. Sometimes I get stuck in a bubble of thinking that the math we are doing is not universal and that BYU has its own way of doing proofs and math and such. However, attending this lecture opens my mind up to the world that mathematics and the way of proving them are universal. While I could not really follow what was going on, I could see him write down symbols and steps of the proof that I did understand and that I have used before and that was really cool. I am glad I attended.

7.4, due on Monday, March 14

1. The hardest part of the reading is probably understanding the new principle of an automorphisms. Everything else in this reading we have done, just with rings so it is pretty easy to understand. But an automorphism is something that is new, and it is used in the homework problem number 19, and I do not know how to do it. Like what is the point of an automorphism and an inner automorphism of G induced by c. It is so confusing.

2. The most interesting part of this reading was that isomorphisms and homomorphisms have come back and can be applied to groups. This is a really cool connection. I like learning things that I already know. It makes me feel smart, and it makes me feel like I can actually do the mathematics that is in this course.

7.3, due on Friday, March 11

1. The most difficult part of this reading was realizing that because I am still confused on an order of an element, I cannot understand subgroups, especially cyclic subgroups. And the biggest problem is is that I am not sure what it is about a, a^2, a^3,... elements and groups that I do not understand. But I cannot do any of the homework do tomorrow so clearly I do not understand something, and thus, I cannot understand cyclic subgroups and how they work and how to use them, so this is the hardest part of the reading.

2. The most interesting part of this reading was that once again a sub-"something" with only a couple of conditions was introduced. We have subsets, subrings, subfields, etc. and now we have subgroups! It is so interesting how everything in this class connects and parallels to each other. For example, eventually in this section we get to the only conditions for something to be a subgroup is to be a nonempty, finite subset and to be closed under the operation of the group that it is a subset under. Then it is a subgroup! Fascinating!

7.2, due on Wednesday, March 9

1. The hardest part about this reading was understanding the purpose and proof from Theorem 7.8. It kind of makes sense, but I do not really understand the purpose in knowing about an element in a group and the order of that element and the order of that element raised to an exponent, etc. So it makes it really hard to comprehend because I do not know where the book is going with the information in this theorem.

2. The most interesting part of this reading is tightly connected to the hardest part of this reading -- the order of an element of a group. This is so interesting. I would not have expected us to look at the order of an individual element of a group. It is so cool how everything in this course expands itself!

7.1 Part 1, due on Friday, March 4

1. The most difficult part of this reading is going to be understanding how to keep the rules for what is in a group G and what the rules for the binary operation of G are. After I read the section, I skimmed over the homework problems, and with problems like 4 and others, I found myself getting confused with the rules for what goes in the group as an element and then what the binary operation * is. That will be the most difficult part with this new idea of groups.

2. The most interesting part of this section is that I have seen all this before - I just cannot remember where! Maybe we did stuff like this in linear algebra... or calculus... or statistics... or math 300. I actually cannot remember, but as I was reading, I felt like I already knew how to do it all. So I know this first look at groups connects with math I have learned before, I just cannot remember where. But I do think it is cool because then it makes it feel like learning about groups will be knowledge that will be very attainable, which I really think is cool.

6.3, due on Wednesday, March 2

1. The most difficult part of this reading was understanding how the term prime can be applied to ideals. I do not understand how an ideal is prime when if b*c is in it then either b is in it or c is in it. Why does that make it prime? Then there is a theorem that says that something is a maximal ideal iff the quotient ring is a field, and yet on our homework it asks for the maximal ideal of rings that are not fields, so how does that work? I am so confused!

2. This reading connects to ideals to primes, which is really cool (once I am able to understand it!). I have only seen primes as associated with one number, but now we are calling an entire ideal prime! It is weird and confusing but still cool at the same time. I wonder if we will continue as the sections go on to add prime properties to things such as ideals and other topics in this abstract algebra course.

6.2 Part 2, due on Monday, February 26

1. The most difficult part of this section of the reading is keeping all the theorems and ideas straight with the kernels and quotient rings and regular rings and ideals and etc. There is so much to remember to then apply it to problems that are on the homework. I am so confused about which one of those terms when paired with another makes a function isomorphic or homomorphic (like theorems 6.12 and 6.13). It is all so unclear and the proofs do not help me understand them so I just keep getting more confused.

2. The most interesting part of this section is how kernels are being used in theorems to connect our rings back to homomorphic and isomorphic. I would have never guessed that the kernels we saw in linear algebra would come back and be applied to something we did at the beginning of the semester in Abstract Algebra. I love how connected mathematics is!

6.2 Part 1, due on Friday, February 25

1. I am still struggling with finding the cosets and how many there are and what they are etc., so I think I am going to struggle with this section because it deals more with cosets and the multiplication of and addition of them, but I can't even find them to use them yet. So there is no way that I can do addition and multiplication tables of cosets when I can't even figure out what the cosets are. So this is the most difficult part of this section -- the fact that cosets still make no sense to me.

2. It is interesting that R/I is a ring. I had no idea from the last section that this would be coming. I can see how a coset is like a congruence class, but to have R/I - the set of cosets - be a ring, well, that is just really cool. I like seeing how things that make no sense in one section (like R/I) are connecting to a big principle (like rings) that we have been working on all semester. I like when weird things turn out to have a purpose. It is cool.

6.1 Part 2, due on Wednesday, February 23

1. I do not understand what a coset is. It does not make sense to me. I do not see why we need to write a congruence class as a + I for it to be the formal symbol. It is just confusing me.

2. It is interesting that all the properties with mods and being an equivalence relation are applied to an ideal. I figured that would have been trivial and obvious. So it is interesting that the book goes through specific theorems and proofs to show how things work in mod I.

6.1, due on Tuesday, February 22

1. The most difficult part of this reading was that I have never even heard of an ideal before. Most of the time the new terms in this book I have at least heard of so I know that they are important. I have never heard of an ideal so I have no idea where learning about the concept of ideals will take us in this course, so this makes it conceptually very difficult for me to learn about them when I do not know what they purpose it.

2. The most interesting thing in this section was how in order to prove if something is an ideal, you use similar properties as you would in proving something is a subring. That is interesting because when I first read the definition of an ideal, I was not thinking that is was similar to a subring, so I like that there are connections with these new ideas and theorems and that some things in math are similar.

5.3, due on Friday, February 18

1. The hardest part of this section is understanding how to use the principles and theorems and apply them to actually proving if something is a field. I was starting the homework for this section and a lot of the problems ask you to prove is something is a field. I did not know how to apply what I had just read with the theorems to be able to compute a proof to show is something is a field. So this is the most difficult part of the reading.

2. I think it is really cool that Theorem 5.11 helps to explain problems and doubts that people had back in history with complex numbers and negative numbers. It is hard for me to connect Abstract Algebra with normal, useful math. I don't feel like it has a place in mathematics. However, this section showed me how Abstract Algebra is used to explain simple math concepts like negative and complex numbers -- things that would not be as common had there not been Abstract Algebra and this theorem to help people grasp the concepts.

5.1, due on Monday, February 14

1. The most difficult part of this reading is not the new theorems and properties, because they are things we have already seen before with mods just applied to functions and polynomials. The most difficult part of this section came from looking at the homework problems and realizing that I have to remember all the other theorems and properties and concepts we have discussed up until now and apply it to this new twist on the topic (like having to remember how to list different congruence classes and apply that to polynomials and the properties of the nonzero constant polynomial). That is hard because I do not remember things or memorize quickly. I have to look in the book and there are too many sections to do this to remember things to use for the homework. So this is the most difficult part.

2. The most interesting part of this reading is seeing mods and congruence classes applied to polynomials and fields. I would have never seen this coming, but I guess this has happened a lot in this course - the building on of the same theorems applied to different things - that I should have expected polynomials to come back into congruence classes.

9.4, due on Friday, February 11

1. The most difficult part of the reading was keeping straight all the new notions, with things like [a,b], (a,b), ~, and a/b. I am really bad with new notation and remembering what it all means and why, so that is the most difficult part of this new section.

2. It is interesting that this new material uses the terms of an integral domain and such, which I have never seen before this class, and connect them with equivalence relations, and the "~" symbol and meaning because we talked about that stuff in Math 190. But it has been awhile since I've had that class, so I don't remember a lot, but I do remember we did stuff with it, so it is neat that it connects to Abstract Algebra and integral domains.

Review Questions, due on Wednesday, February 9

1. Which topics and theorems do you think are the most important out of those we have studied?
I think the topics of rings, homomorphisms, isomorphisms, and fields are most important. So knowing the definitions and important theorems and properties about each of these topics is probably really important. It is also probably good to know about the Euclidean algorithm.

2. What kinds of questions do you expect to see on the exam?
I expect to see questions that require you to know the definitions and use them to prove theorems similar to ones we did in class and on the homework. I expect the questions that are more algebraic in nature to be like the ones we did on the homework.

3. What do you need to work on understanding better before the exam?
I just need to review all the definitions. I understood the material when I was doing the homework, but it is recalling it on the spot and without notes that is difficult. So just reviewing all the elements that make something a ring, integral domain, field, etc. is what I need to study for the exam.

4.4, due on February 7

1. The most difficult part of the reading is how to determine how to find a root for really hard polynomials and/or mods. The roots are the biggest part of this section and the theorems make sense, but applying how to figure out if something is a root for more difficult problems is the hardest part of this section.

2. The most interesting part of this reading is how something as simple as roots, which we learned about in middle school while taking algebra, are being used and applied in this advanced math course. This connection of math principles fascinates me that simple math things can still be used in more advanced courses.

4.3, due on Friday, February 4

1. The most difficult part about this reading was what exactly some of the terms meant because I really had never seen them before or some of them at least I felt were not adequately defined. For example, a nonzero constant polynomial (or a unit) is really confusing to me. I mean, I get what it is, but I had a hard time recognizing them and doing the homework properly because I don't really get what units are in polynomials. The book's definition wasn't good enough. Also, on the homework, we are to prove something about a unique monic associate, and that is in section 4.3, and I have no idea what that is. I used context clues from the problem 1 and its answer, but I still don't understand it enough to use it in a proof.

2. I really like Theorem 4.11 is cool because I love theorems that consistent of 3 or more equivalent statements. Those types of theorems are cool and make writing proofs fun and easy because you can apply whichever equivalent statement fits the proof best. So I like adding another group of equivalent statements to my mathematics knowledge.

4.2, due on Wednesday, February 2

1. This section seems pretty straight forward because it is all definitions and theorems we have seen before, just now they are applied to polynomials. The most difficult part of all of this is how all of them state that f(x) and g(x) need to be in field F. This might be a dumb question and should have probably been cleared up a long time ago, but I do not understand the need or significance or purpose behind the field F and what it means and why it matters. I am not seeing the connection at all.

2. I think it is so interesting how all these definitions and theorems about the gcd and being relatively prime are being applied to polynomials. It is so cool! Also, it is cool how a divisor times any nonzero integer still divides the polynomial. I love unique properties like that!

4.1, due on Monday, January 31

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was understanding the proof of Theorem 4.4. There are just too many new definitions and properties and such in this section that make the proof really hard to understand because the new material with definitions and such have not sunk in yet. It is a proof that I am going to have to go over multiple times before I can really get it.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
I don't know why, but I have always found polynomials fascinating. So I am excited that we are applying ring stuff to polynomials. I just think they are so fun!

Random Questions, due on Friday, January 28

1. How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I have spent between 3-5 hours on each homework assignment. Sometimes it is frustrating because I feel like a lot of the problems are repeats, or like all the parts aren't necessary to get the points across. Sometimes it could be shortened. Lecture and the reading do prepare me, but the problem is in lecture we go over exactly what is in the reading. We go over the exact proofs, examples, etc. that are in the book. I know how to read a book, so if lecture focused on more application of the theorems instead of repeating what the book said, then I would be able to be more prepared for the homework.

2. What has contributed most to your learning in this class thus far?
I have a clear understanding of the material because lecture is so clear. That really helps. You are one of the best professors I have had because you speak and write clearly, and you lecture at an appropriate pace and at an appropriate level of knowledge. I feel like I can really learn because it isn't over my head or unclear. I really appreciate that, and for me, how the professor teaches has a huge impact on my learning. So thank you.

3. What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
I think including in lecture more application of the theorems and definitions and stuff would be really effective because homework is a giant mathematical leap from the reading and the lecture. To bridge the gap, help us in lecture to understand application principles involved in the material to prepare us for the hard homework.

3.3, due on Wednesday, January 26

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was following how the proofs work for some of the corollaries and theorem sthat deal with homomorphism and isomorphism. For example, I have a hard time connecting Corollary 3.13 with the work we did with subrings from the previous sections. There are too many definitions, rules, and theorems to keep up with to understand all these weird properties for homomorphism and isomorphism.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
I think it is interesting how functions have popped back into this mathematics. I would not have expected functions to be included in working with rings or with our new concepts of homomorphism and isomorphism. It is interesting how most mathematics relates to all other parts of mathematics as well. Crazy!

3.1 Part 2, due on Friday, January 20

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material was knowing HOW to prove that all the axioms are true for something to be a subring. Most of the examples said this is, this isn't and because of what axiom is false. But nothing took you through the process of figuring out how to prove which axioms are true and false. I have a hard time coming up with proofs on my own and knowing if they are efficient or not, so that is the hardest part of this reading for me.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of this reading is how matrices are being used a rings and subrings. It is connecting to what I learned about matrices in linear algebra. I kind of thought I was not going to see matrices again, so this is interesting -- using them to learn and think about rings and subrings.

3.1 Part 1, due on Wednesday, January 19

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was some of the technical details with the new definitions. For example, a couple of them talk about 1-sub-R not equaling 0-sub-R and other equations equaling 0-sub-R. I do not understand the 1 not equaling 0 or the other equations equaling 0 especially because I do not know what the sub-R's after all these mean. I am so confused and I have no idea where that technicality came from or what it means.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was how you could make a ring out of a set of letters and you could even redefine what addition and multiplication are, in particular stuff like homework problem 11. I had never thought about doing this type of math with letters and with new definitions of addition, multiplication, etc. It is cool, and I think it makes it applicable to then apply this stuff to other types of numbers if you can do it with letters.

2.3, due on January 14

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was trying to follow the proof the book gave for showing that point 3 in Theorem 2.8 was equivalent to point 1 in Theorem in 2.8. I got very lost when they assumed that when a*b=0 in Zp, then a=0 or b=0 and used it to show that p is prime. There was too much assumptions done throughout the proof for me to follow, so I am not yet convinced of this truth.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.11. I think it is so interesting that if d/b then the equation has solutions, and furthermore, that the equation has d distinct solutions. I liked being able to use this theorem to do the homework. It was so simple. However, I am interested in seeing the proof for why the equation has d distinct solutions. I would be very interested to see how that works.

2.2, due on January 12

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the material for me was how in the previous section we had been dealing with mods and now we are dealing with Z and my mind gets confused on transferring what I had learned about mods over to calling it all Z's. I mean, I think all the Z stuff is like using the idea of a mod or a change of base, but I'm still confused when approaching problems like the ones on the homework.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The most interesting part of the material was Theorem 2.7 where there were like 10 properties that hold true for classes in Zn. I think that is fascinating how there are all these properties - very cool. Also, the proofs for them seem simple enough that it is interesting to see how they all simply work.

2.1, due on January 10

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the reading was understanding the corollary about how [a] and [b] of the same mod can either be disjoint or equal and nothing else. I do not believe the proof and how it works; it does not convince me.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
The mods connect to what I learned in Math 300 and Math 290. It also relates to change of bases; the process is used the same way.

1.1-1.3, due on January 7

1. (Difficult) "What was the most difficult part of the material for you?"
The most difficult part of the reading was about how to apply the division algorithm. I understand what that theorem states, and I can follow the proof fairly sufficiently, but I do not understand how to apply it. When using just variables when applying it to other proofs, conjectures, etc. I get really lost on how to use the algorithm to prove something else when all the numbers are just variables. I get very lost.

2. (Reflective) Write something reflective about the reading. This could be the answer to the question "What was the most interesting part of the material?" or "How does this material connect to something else you have learned in mathematics?" or "How is this material useful/relevant to your intellectual or career interests?" or something else.
This material connects to some of the math material I studied in Math 300. I remember studying the properties and theorems of primes and relative primes. For example, we learned about the fundamental theorem of arithmetic and what it means to be relatively prime. We studied the theorems in relation to what Euclid did. It is interesting to then see some of what he did and what we studied appear in Abstract Algebra. I like being able to build on what we studied in History of Math in Abstract Algebra.

Introduction, due on January 7

1. What is your year in school and major?
I am a junior double majoring in Math Education and Theatre Studies.

2. Which post-calculus math courses have you taken? (Use names or BYU course numbers.)
Math 313, Math 314, Math 334, Math 290, Math 341, and Math 300

3. Why are you taking this class? (Be specific.)
I am required to take this class for a Math Ed degree. I honestly do not know why this class is required for that degree. I do not know how Abstract Algebra can help me teach math (particularly to middle school students because that is the level I want to teach). However, I hope that clear connections will be made in this class to the fundamental math that I will be teaching in schools. But honestly, I would not be taking this class except that it is required, and I want to get a degree in Math Education.

4. 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?
My Math 290 professor was the least effective math teacher I have ever had. Firstly, I could not follow anything that he said. He was very unclear with what he was talking about and where his lectures were going, and he was very hard to understand. I like to have a clear layout of where a lecture is going and a clear layout of how what we are learning connects to prior mathematical knowledge. I also like to have a relationship with my professors, and I like to go to office hours. He always made me feel stupid and uncomfortable, and for me, that is something that really inhibits me as a learner. If a professor does that, I lose trust in him/her, and I lose confidence in myself, and then I do not learn as much or do as well. He basically made me feel like I could not do it, which led me to believe that I could not do it, and that just led it to become more true.

5. Write something interesting or unique about yourself.
I love theatre and, in particular, musicals. I could talk about musicals for hours. I also love to act, and I am acting in a mainstage BYU production this February.

6. If you are unable to come to my scheduled office hours, what times would work for you?
I am not able to come to your office hours. The following times would work: 1-2pm MWF, 3-4pm MWF, and 2-4pm TTH.