8.5, due on December 7

I felt like this section was easier than some of the others recently (it helped that a page and a half of the section was describing dihedral groups that we have already learned about). I didn't understand the first proof (theorem 8.27). I didn't know how they obtained the equation they call the class equation and I was unsure if the Ci's were just divisors of G or what they were. I also didn't really understand the proof of theorem 8.34. It was really long and I wasn't exactly sure how they used all these assumptions to construct the multiplication table. I would be helpful to go through that proof again.

I had one other question which was about the group Q, the quaternion group. I missed one or two days and I'm wondering if we talked about it on one of those days. I don't know what it is or why it exists and has no similar groups in other orders except of order 12 (is the reason there is a T group similar to the reason there is a Q group? It was a little disheartening that the chapter ends by telling us that there are 267 different groups of order 64 and that there was no way to know the number of distinct groups of order n. How do mathematicians find the maximum number of distinct groups for very large groups?

Looking over the study exam for the final made me feel a little overwhelmed. I feel like it is all important and that means I have a lot of stuff to memorize. I really need to review the stuff on rings because it has been a while since we covered that material. I also don't fully feel comfortable using the theorems from this chapter to prove things. I want to go over the practice problems from the study guide especially numbers 8, 9, 12, 17, and 18.

I have learned so much from this class. I don't know what will be specifically important to me in my future because I don't know what I want to do with my math degree. But I think that the improvement that I have made in being able to prove things will be really helpful in future classes, especially theory of analysis which freaked me out because I couldn't remember how to do proofs at all.

8.3/8.4, due on December 5

By far these were the most confusing sections of the class. I don't understand what is going on with the Sylow second theorem. I don't even know what they mean by P=x^-1Kx because I thought that doing x^-1Kx was used to determine if K is normal in G. What would it look like to have two sylow p-subgroups that were equal in this way? But my level of confusion really grew in section 8.4. I can read theorems and think I understand them, but then reading the proofs really showed my that I did not. I was so confused by the use of the conjugacy classes as used in the proofs. I also was getting mixed up and confused by the many uses of Lagrange's theorem. I need to reread these sections again and I need to go through the proofs in class really slowly.

I thought that the most interesting part of the section was talking about conjugacy classes and the centralizer. Throughout the semester as I have looked up proofs and information for clarification, conjugacy and centralizer have come up really frequently. People are always using them as the easiest way to prove things that we learned earlier on (which is obviously unhelpful at the time because we hadn't learned it). I am happy to finally be talking about them because (once I funny understand them) I think it will make a lot of aspects of group theory more clear - or at least I'll be able to understand some more of the online discussions and proofs of group theory.

8.1, due November 30

My mind was overwhelmed with this section. It was interesting but I'm not sure what I read. I was really confused by the proof of theorem 8.1. In fact I had no idea what was going on to prove that f is a homomorphism. I would also like to go over the general idea on which the whole section is based which is writing the elements of a group as sums or products of elements of two subgroups. I don't really understand why that works.

I think it is interesting to look at the problem of what a group is isomorphic to by crossing some of its normal subgroups. How do we know which normal subgroups to use? Is it based on the size of the subgroup or is it trial and error?

Test Review, due on November 27

I think some of the most important theorems and topics we have covered are related to homomorphisms and isomorphisms of groups. I feel like it is easiest to interpret groups when we see what they are isomorphic to. Thus theorems that tell us about normal subgroups and give us the ability to use the first, second, or third isomorphism theorems are the most helpful to understanding groups. I could see this test being very similar to the last one which asks us true false questions about whether certain sets are groups, certain types of groups, whether groups have elements of certain order, etc. Then there will be some questions asking for examples of groups that are isomorphic to Zn or ZnxZm and examples of other types of groups. Then there will probably be some large proof problem asking for some use of the First Isomorphism Theorem for groups like the last test had for rings. I think that there will also be some problem asking for a proof about the order of elements in a group and proving what those orders are (so using Lagrange's theorem).

I need to work on a lot for this test. I think I understood a lot of the material but I didn't really internalize it yet. I also need to work on the last couple of sections that we covered. I have the most difficult time with the material regarding Sn groups (like talking about permutations being odd or even and talking about groups An). I would like to go over problems 7 and 8 on the practice exam because I have a difficult time finding subgroups of groups and I am wondering if there is an easier way to do it that to just check a lot of options.

7.10, due on November 22

I would honestly say that this section is probably one of the most difficult for me to understand. I was glad it was so short because I don't think I got very much out of it and if there had been more information that would have been more information that went over my head. I think I am having a difficult time understanding what An is. I don't know what a group like this really is. I also didn't understand why lemma 7.53 is true. Nor did I understand case 3 of the proof of theorem 7.52 or actually why we could boil down the proof to three cases in the first place.

It is interesting to me whenever there is a theorem that is true for all n (here just n not 4). It is especially interesting when these proofs aren't by induction and are relatively short in length. Most proofs of theorems with that broad of an assumption turn out to be more difficult that just three cases like the one for theorem 7.52.

I had an additional question not on this section but on the homework for section 7.8. I won't be able to go to office hours since I have a test to take so I was hoping if there is time you could address it in class or I will ask you after class. On problem 1d) I can't figure out how to interpret the function. I guess f in Sn looks like some cycle (12...n) or something. But then I thought (k) must be one number inside parenthesis since 1 less than or equal to k less than n. So is (k) just the identity because isn't that how we write the identity in cycle notation? (1)=(2)=(3)=(4)=e in Sn? So then what is f(k)? Wouldn't that always be just f? And is the function just plugging in different values of f (different elements of Sn) or also different (k) (which would all be the same thing if my above assumption is correct)? Please help.
                                                                         

7.8, due on Novermber 21

I feel similarly about this section as I did about the section on the First Isomorphism Theorem for rings. It was interesting, but the proofs were long and confusing. I didn't understand the proof of theorem 7.45 and since the proof of the First Isomophism Theorem for rings confused me the first time, I would love to run through the proof of the First Isomorphism Theorem for Groups in full instead of just saying "translate the proof of theorem 6.13".

I really liked the Third Isomorphism Theorem for Groups. I thought it was interesting that the book defined another isomorphism theorem in the section rather that just having us discover it in the practices (like in the section about rings). Is that because this theorem is more useful in groups than in rings?

7.7, due on November 18

The most difficult part of this section for me to understand was actually one of the examples. The second example on page 218 was confusing because they did the arithmetic for the example but didn't do any of the verification that G/N is cyclic and of order 4. I think that I am confused as to what a cyclic quotient group looks like and how to use a multiplication table to determine that. If I could get to the determination that it had an element of order four then I would know that it was cyclic and that it was isomorphic to Z4. But I can't figure out how to tell from the table that something has order 4.

I thought the most interesting part of the section was the last theorem where it is shown that if G/C is cyclic then G is abelian. Do we know in what cases G/C is cyclic? Does it happen often? It would be nice to look at examples where this theorem is true. I would also like to look at examples where G is abelian but G/C is not cyclic.

half of 7.6, due on November 16

I am really glad that we are going through the proof of theorem 7.34 tomorrow in class because I didn't quite catch what was going on. I also didn't really understand the proof of theorem 7.33. I would like to go through that again if possible.

I was wondering when we would get into quotient groups so I suppose it's good that we've got the background for those now. I don't think that I really grasp how normal subgroups have similar properties to ideals. I really want to take a class on the application of rings and groups because I find them interesting but sometimes too abstract to grasp without application.

half of 7.5 and half of 7.6, due on November 14

I thought the rest of section 7.5 was mostly understandable. What I read from section 7.6 was very confusing though. I'm sure missing class on Friday is a main reason for my confusion. I am having a really hard time conceptualizing what a congruence class in a group is. So talking about left congruence classes now is increasing my confusion. I think that going through examples in class will really be the only way for me to grasp this.

I really enjoy being able to classify finite groups as being isomorphic to some particular modulo of the integers. I feel like isomorphism in groups seems to be much more easy to identify than in rings and thus I feel like I know so much more about a finite group when I see it than I did about a ring. I actually enjoy rings more I think. Maybe that's just because I'm having a harder time understanding groups.

half of 7.5, due on November 11

The most difficult part of the reading to understand was corollary 7.27 and it's proof. Maybe I had been reading too long but it didn't make sense to me. I also didn't quite get the proof of theorem 7.25. All in all though this section was easier to understand than most.

I thought Lagrange's theorem was really interesting. The proof was relatively easy to understand so it makes me believe that it has been developed over time from an original much more complicated proof by Lagrange. He must've done much more work in group theory developing more than just that one theory or else I feel like someone else would have figured out this result sooner.

7.9, due on November 9

I didn't understand the whole second half of the section starting with transpositions. I guess what was the source of my confusion was starting with how (12)(12)=(1). I think I might be just not understanding the notation. After that, I couldn't follow the whole odd or even number of transpositions proofs and the last theorem 7.51 was the most confusing. I did not understand the definition of An and the proof that followed.

I thought the most interesting part of the section was that we developed the notation for symmetric groups to simplify problems. It seems like a really clever and simple solution to writing the long form of elements in a symmetric group. I also thought it was interesting that theorem 7.51 was proved using the First Isomorphism Theorem. I want to know how this theorem is proved when you learn about group theory before ring theory and don't have the First Isomorphism Theorem to use.

7.4, due on November 7

The ideas in the section made a lot of sense to me because we already have some familiarity with isomorphisms and homomorphisms. However, I had a difficult time following the proofs of the section. Specifically, I didn't understand the reason why we chose the function f(i)=a^i for theorem 7.18. I also didn't understand part 2 of the proof of theorem 7.19. And then the entirety of Theorem 7.20 was completely confusing and then of course the proof of corollary 7.21 plays off of that proof so I didn't get that one either.

I thought the most interesting part of the section was how many isomorphisms we can guarantee for groups. We know there is always an isomorphism from a finite group to a subgroup of the symmetric group. And we know that every group is isomorphic to a group of permutations and cyclic groups are isomorphic to the integers or the integers mod n. It seems like it is relatively easier to find isomorphisms for groups than for rings but maybe that's not true.

7.3, due on November 4

I don't know what it was about this section that really got to me, but I had very little understanding of this section. After theorem 7.11, the rest of the section went over my head. I need to go through the whole section again in class with lots of examples and then reread the section myself. I really don't understand what I read today.

I thought it was interesting that the proof of theorem 7.10 was so short. It seemed like we could almost just assume that H is a group. As interesting as this was though, I was confused by the fact that we could assume (ii) to prove the theorem. I thought assuming the theorem was never considered a proof.

7.2, due on November 2

For the most part I followed this section well. The most confusing parts were parts 3 and 4 of theorem 7.8 and their proofs, and the proof of corollary 7.9. I understood what corollary 7.9 was saying but I couldn't follow the proof (probably because I didn't the references of how to use theorem 7.8 part 4.

I was glad we had already spent a few minutes of class discussing the order of an element because I don't know that I would have understood the discussion without having already gone through some examples. I'm curious to know how order plays a role in groups. Like it was mentioned in class, we will be using it probably in conjunction with isomorphism between groups. Because of theorem 7.8 part 3 where a^i=a^j iff i is congruent to j mod n, I'm wondering if we'll be making congruence classes in groups and using those in different ways. I can see how this chapter could go from relatively easy to understand to completely abstract relatively quickly.

7.1 Part 2, due October 31

The most difficult part of the rest of this section was just wrapping my mind around  two of the examples. In the example on the top of page 170 I don't think I understand why the set of these matrices is a group only if ad-bc is not 0. I also couldn't quite wrap my mind around the Cartesian product of a group of number and a dihedral group.

So far I am enjoying groups. They seem to mostly just be classifications of things that I already knew about. When you said on Friday that we would be learning about dihedral groups on Monday I was a little freaked out. I was pleasantly pleased that dihedral groups are just rotations and translations of polygons on a plane, which is simple enough.

Half of 7.1, due on October 28

Several times during the semester, I have been looking online for explanations to topics we are covering only to read proofs and listen to explanations of rings using group theory. They will say something like "a ring is just an abelian group with these extra properties" or "we can tell that such and such is an abelian group and thus..." Finally after reading the first part of this section I understand a little bit better what they were talking about and I am glad that the axioms of a group are fewer in number and as easy to understand as ring properties.

I understood most of the reading, which was a nice change from the last chapter. It was a little confusing from the diagrams at first that we do the right function first and then the left function but it makes sense when you think about it. The only other thing that got me was the symmetric group and the order of Sn. I think I misunderstood that part or just plain didn't understand it.

Review for Exam 2, due on October 26

I think that the First Isomorphism Theorem for rings is one of the most important things we have learned (because it was stated in class that the proof essentially used all the skills we have learned). I also think that knowing all of the definitions is going to be huge. I would expect (if the test is formatted similarly to exam 1) that there will be five or so definitions probably about kernels, maximal ideals, principal ideals, homomorphism/isomorphism of rings, and quotient rings; five or so questions asking examples of ideals that are non-prime and maximal, some type of polynomial, examples of fields that properly contain the rations and reals, and rings with some special property of their quotient ring; two or three questions that use a lot of the theorems to prove and a the proof of the First Isomorphism Theorem for rings. 

In addition to wanting to see numbers 4, 5, 7, and 9 from the practice exam (or all of them if you are so inclined), I would also mainly like to see examples of every variety of thing we have covered. I know this is an unrealistic request but I think the reason why I don't understand what we are learning is because I don't have enough examples to understand the definitions from and I am a learner by following examples. 

6.3, due on October 24

I was very confused by the several of the proofs in this section. And actually by reasoning through a bunch of the terms in this section too. The whole concept of a maximal ideal is confusing to me and I couldn't follow the proof. Another thing I was confused with was an example of a prime ideal on page 6. They used the zero ideal as an example but I thought part of the definition was that p couldn't be 0 (earlier on the page it says "if p not 0 or plus minus one, then p is prime if and only if whenever...). I feel like this is all one level too abstract for me to wrap my head around (not that I don't find it interesting when I do understand what is going on).

I guess something interesting about the chapter is that a bunch of the theorems are if and only if which might imply that we could use some of the conditions to determine is p is a prime number. I know that is something that is difficult to do with very large numbers so is examining ring structures a helpful way to find primes? For example, if we took R/P for some number and found that it was an integral domain, then we could know that p is a prime (or irreducible if we are working with polynomials) by theorem 6.14, right? Or maybe I'm misinterpreting the theorems which would be a good thing to clear up.

6.1/6.2, due on October 19

The most confusing part about the rest of section 6.1 was the notation note about the plus sign not meaning addition in congruence classes modulo I. I don't really think I understand what is is contained in those congruence classes. Is a+I a plus any possible i in I and that's why we write it in that form? Or is a+i a different congruence class than a+j for i,j in I, but we would write them both as a+I? Section 6.2 was a bit more confusing. It's a little strange that (a+I)+(c+I)=(a+c)+I. Why wouldn't this be plus 2I? Although I guess I wouldn't know what 2I really meant. How many congruence classes are there in R/I? Can we do an example of an Ideal and show all or most of it's congruence classes? The most confusing part of 6.2 has to be the First Isomorphism Theorem and it's proof. I thought I understood the discussion following the theorem but then the proof went over my head and the examples didn't really help me. It would be nice to discuss this part of the section.

Despite not really understanding how the First Isomorophism Theorem works, I think that is is really interesting that we can find always find a ring isomorphic to R as long as we have a surjective homomorphism between R and another ring. Suddenly it makes sense why we were concerned with surjective homomorphisms in an earlier chapter. I kept thinking that they weren't really very important because isomorphisms are so much more revealing. I am curious to know some of information that we can determine about R when given a homomorphism.

half of 6.1, due on October 17

After the preface in class that this chapter would be the most confusing information covered this semester, I was nervous for this section. However I think I actually understood a similar proportion to what I have understood from reading other sections. It wasn't until the end of the reading that I began to get confused. I'm not sure I understood the example given about not being a principle ideal on page 137. I just didn't really grasp the situation. The I thought I understood the finitely generated ideals until it said that the generators of the finitely generated ideal need not be unique and I read the example. I think out of the whole section that part was the most confusing. I would like several examples of this type of ideal.

It feels good to have a name for the set of  multiples of a modulo since I knew that there must be some way to categorize them and their importance. I am curious to know what the importance of ideals is. They are interesting in that they are closed under multiplication but I don't really know what importance they will serve. It seems like another level of abstractness to keep track of but I can't connect it to any concrete application to help me keep it straight in my mind.

Talk by Mark Embry, from October 13

I am actually enrolled in the careers in math lecture series but this was the first class period during which I felt that the presentation merited a blog post. Additionally, I found it very interesting. This lecture was one of the most interesting lectures I have attended. Particularly because I have had few exposures to applications of the mathematics covered in most of my classes. I thought the most interesting part was how graph theory and matrix theory are so connected. I am also in graph theory right now and we have talked about creating the adjacency matrices for the purpose of finding isomorphisms between graphs. It was enjoyable to see how the matrices serve so much greater purposes in analyzing data and graphs.

I think the most difficult part of the lecture to understand is when he was discussing ranking the football teams. He said something about a theorem called Perron Frobenius Theorem at which point he repeated to perform some same process to the data repeatedly in order to finally come to the eigenvector. My linear algebra is very rusty so perhaps this would make more sense if I had just taken the class or had more practice. There were, of course, several other places in the discussion that I did not understand because I have not studied enough linear algebra. But in all, I was pleasantly surprised that I could follow the majority of the discussion.

5.3, due on October 14

I think I generally understood this section. The most difficult part was keeping straight in my mind what ring I am in. When I'm going from R to R[x] then to R[x]/(p(x)) then to doing congruence classes arithmetic in K. I just get a little jumbled up when reading examples so it would be nice to do a couple of more in class. I guess I also need a little clarification. Are we essentially saying that every polynomial must have a root therefore we will create larger rings specifically so that they contain this root when the polynomial was irreducible in the original field.

I was pretty excited to learn how the construction of the complex numbers works. It has always been difficult for me to understand what they are and were they came from. In fact the other day I was tutoring my little sister who is in her first semester of high school algebra. She was covering the types of number systems and I was having a difficult time telling her that the real numbers weren't really all the numbers because I couldn't explain where the complex numbers came from. She was confused and I was confused because I didn't have the background knowledge to comfortably tell her the reason. This goes back to the exact thing we were talking about in class where it is important for a teacher to have a deeper understanding of the topic than the student.

5.2, due on October 12

I think the most difficult part of this section was when it started discussing constructing a ring by identifying a  F with it's copy F* inside F[x]/(p(x)). I don't know what identifying means. I don't know how they got from one thing to another in that example on page 126 with the addition and multiplication tables. I also didn't understand why x^2-2 and 2x+5 are relatively prime as in the example on page 127 and how I should know this based off of information from the top of my head. I think I'm starting to fall behind in my understanding of the things we are discussing and each section is getting more and more confusing for me. I need to reread everything from chapter 4 I think.

I don't understand how they constructed a ring containing a subset isomorphic to Z2 but I think it is very interesting that that can be done with any field and a nonconstant polynomial. I am also relieved that addition and multiplication work the same way for congruence classes in polynomial rings. I would have dreaded it if they had had special rules.

5.1, due on October 10

I found this section to be easier to understand than the previous several sections for the most part. That being said, corollary 5.5, it's proof, and the following examples for it were rather confusing. I think I understand what the corollary is trying to say because I know the rule for the number of congruence classes in Z, but I don't understand how they word the description of the number and distinctness of the congruence classes. I don't understand how the congruence classes would be identified given a modulo because I don't feel like it's evident given the examples what all the possible remainders could be from dividing polynomials.

I'm curious to know the applicability of congruence classes of polynomials. It's interesting that congruence works the same in the ring of polynomials because it's a field. Then does that mean all fields have congruence modulo something? If so, are any of them important to look at? If they aren't important, what makes modulo integers and modulo polynomials more useful? These might be stupid questions but I feel like all of the discussion about congruence is something so interested but something I have never thought about or studied before and I feel like there must be a lot of applications or at least a few really important applications.

4.5-4.6, due on October 7

I felt like I was understanding most things about irreducibility until halfway through section 4.5. And then they brought in prime numbers as criteria for reducibility and I got completely lost. I read the rest of the section and did not understand hardly any of the proofs. Since I am a visual learner it will be really helpful for me to see examples of how to use the criteria mentioned in the section to figure out if a polynomial is irreducible. Unfortunately, it got worse as I read section 4.6. Beginning with Lemma 4.28 and the new notation they introduce I essentially didn't understand any of of the section. I really cannot understand the notation and it doesn't help that I am unfortunately relatively unfamiliar with complex numbers.

Because I was so confused, I found it difficult to make a lot of connections between this section and other things that I have learned. I do think it's great though that there are more ways to determine if a polynomial is irreducible because the methods that we were using before are not very efficient in many cases. It is also interesting that there have never been any formulas derived to find the roots of polynomials above fifth degree. My inner desire to solve unsolvable problems makes me want to study this, but knowing that many geniuses have examined this gives me little hope that I would get anywhere.

4.4, due on October 5

I struggled a little bit to understand some of the proofs for the theorems in this section even though the theorems for the most part were things that I am familiar with regarding polynomial functions. I particularly struggled with the proof of corollary 4.16 (especially because proof by induction is one of my weak points). I got lost when the book stated 0F=f(c)=(c-a)g(c). And couldn't follow after that. In the proof of corollary 4.18, I felt like I must have missed something from before because I didn't know that -c^-1d in Fwas a root of cx+d in F[x]. Or maybe I do know that but am just confused about what ring we are in. I also didn't understand the proof of the final corollary. Why would every element of F be a root of the polynomial f(x)-g(x)?

I was so happy to finally talk about functions. Since we had made such a strong distinction between the ways we use the symbol "x", I was wondering how functions were ever going to be tied to polynomials we were using in F[x]. It seems like such a simple description for who we define the polynomial functions that I don't know why I hadn't learned it earlier. What I wonder though, is how the continued separation between the polynomial and polynomial functions will continue to be relevant. In cases where I have applied mathematics, I have never seen a time when it would be useful to think of the polynomial as not a polynomial function. Then again, I probably don't have much experience so I'm sure there are plenty of situations where the ring of polynomials is useful.

4.3, due on October 3

I found a bunch of these proofs difficult to understand just because they stated to rewrite things from previous sections but I learn much better by having proofs written out with the correct wording instead of flipping back and forth trying to figure out what to change here and there. I think I started becoming especially confused at theorem 4.11. I can't understand how condition 3 means that a  polynomial is irreducible. Isn't part 3 saying that it can be factored by two polynomials so therefore it is reducible? The proof didn't really help me understand this any better. The last theorem and proof would be awesome to go through as well because I had a difficult time with that proof in the original section it was introduced.

Like I said previously, I enjoy talking about polynomials and seeing how connected the properties are between them and types of numbers I've studied about all my life. I feel like I never had as much context about polynomials and why they function with arithmetic and have properties that are essentially so similar to all these other sets of numbers. So for this section in particular, I liked finally making the connection that irreducibility is the same as being prime in the integers. That new lens of looking at reducibility helps me understand it better.

4.2, due on September 29

This section got a little weird at the end. It was difficult for me to wrap my mind around the Euclidean Algorithm for polynomials. I can see how I would be able to do that very easily. In fact it seems unless you do the polynomial division twice for each step you won't be able to move to the next equation. Also I was confused by the proof of theorem 4.5. I tried to read it and I think that I can't wrap my head around what it looks like to write a gcd in the form f(x)u(x)+g(x)v(x) and so the proof didn't make sense.

I really enjoyed this section though because I had never thought about treating polynomials like integers with factors, gcds, and linear combinations. I think that it is more fun to do factors of polynomials. I also really thought that it was clever to redefine the gcd to be unique by introducing the idea of a monic polynomial. It should have seemed obvious to me that this would be the solution to infinitely many factors but it was a huge revelation when they told us how to fix this problem. I enjoy how much arithmetic is in the discussion of polynomials.

Test Review, due of September 26

For risk of sounding obvious, I think the most important topics we have covered are rings. What constitutes a ring, types of rings, and special additional qualities rings can have.I really need to do some more review of definitions. I think that is the most vital aspect of this test will be keeping all of the definitions straight in my mind and memorizing examples of different rings.  

I expect to see a bunch of questions mainly about how to show or prove something is or is not a ring or a certain type of ring. I would love to see a proof showing something being a commutative, and integral domain, a field, with identity. Essentially I just need to see another round of proving all of these properties. I am very worried about this exam because I don't think I've committed the necessary items to memory and instead have been using my notes extensively for assistance on homework.

9.4, due on September 23

This section was by far the most difficult to understand. The notation during the majority of the chapter was so confusing. For most of the time I didn't know what was going on. I think where I started getting most confused was when the chapter started talking about integral domains and proving that F is a field. I also don't understand what is being talked about in equivalence classes of quotients. Is that saying that any fraction that can be simplified is in the same equivalence relation with each of it's reduced forms?

The most interesting part of the section was when the book all of a sudden decided to tell me all the confusing notation represents typical division symbols. I was a little perturbed by the fact that the whole chapter hadn't been written using this format because I think I might understand it a little better. I was glad though that this chapter finally covers division so we can begin using that fact.

Reflection, due September 21

I typically spend between 3-4 hours on the homework, although a few of them have take fewer hours (maybe two at the shortest) or more hours (around 5 hours). Typically I know what I'm supposed to do from from the lecture and reading but frequently I just get stuck for an hour or so at a time where I can't figure out how to proceed with the problem. Sometimes I need to look for a hint for the next step when I have been stuck on a problem for more than an hour. Previous to this course, I didn't read the textbooks for my math classes before the lectures. I think attempting to understand the topics before lecture has really helped me to get more out of lectures.

I need to attend the TA hour on Tuesdays because I think I could really benefit from going over my homework problems I missed and have a little assistance with the current homework. Unfortunately, I live some distance away and don't have classes on Tuesdays so I have yet to come to campus for that TA hour. It would also be beneficial for me to get ahead by a homework assignment so I can receive help in advance. I think it could be really helpful to try to go over tips for some of the problems of homework at the beginning of class or ask which homework problem was the most difficult for the majority of the students and show some steps for that problem.

3.3, due on September 19

I felt like the first part of section 3.3 was relatively easy to understand. However, some of the examples began to confuse me because I feel a little shaky on some of my rings like matrices and Cartesian products of congruence classes. I am also somewhat confused by why the homomorphic examples on page 72 are not injective or not surjective (and how to show that they are the other one). I did not understand the definition of image and think that an example would clarify that for me. Lastly, I was really confused by the example on page 75 about the ring Z8 and why it's not isomorphic of Z4 x Z2.

In graph theory, we are talking about isomorphism so I really enjoyed that the description of isomorphism on the first two pages of this section were the same as how we described it in graph theory (relabeling the elements). I keep having the feeling that somehow these two classes are more connected than I currently understand and that really intrigues me (although maybe they aren't connected at all and it's just wishful thinking that graph theory has meaning). I also really enjoyed that we were shown techniques for finding if rings are isomorphic and for finding that they are not. It's funny that in graph theory it's typically easier to find a way that graphs are nonisomorphic than showing that they are isomorphic whereas in this section it seems like it's easier to find the isomorphism than the difference between the ring. Although possibly that might change with practice.

3.2, due on September 16

I don't quite understand theorem 3.8. Is it saying that for a unit, there are any number of equations that work with ax=b?  ya=b is the example in the book and is that the only other case where an equation of this form works? or could wa=b and ca=b? I don't understand the multiplicative inverse example given in the book where 7^-1=3 and 3^-1=7 in Z10. What does the negative power mean for congruence classes? Lastly I was confused with the discussion on theorem 3.11 which shows that ax=1R for all a in R. But the proof didn't really make sense in my mind.

I likes the proofs of theorem 3.5 because they seemed like a lot of clever little tricks that I might have been able to think of (and it always feels nice to think you might be able to do the work the book did). I'm also happy to finally have subtraction in the toolkit because it is so helpful for proofs. I also thought it was very interesting that all fields are integral domains. But I guess I'm a little confused as to why only finite integral domains are fields. This whole topic is a little mind boggling to me and it's taking a little time for the terminology to sink in. Hopefully as we go, I'll get better at what a ring is instantly and then be able to build off of that better.

2.3, due on September 9

The most difficult part of the section for me was the end of the section. I don't really understand how the book came to the conclusion that if d divides b than [a]x=[b] has solutions in Zn. I also don't know why there would be d number of solutions. I figure that those proofs will be really difficult for me to come up with. I also didn't really understand why it works to change things Zn to Z and what that means (technique i from the basic techniques for proving statements in Zn).

After reading section 2.2 and doing the homework I could tell that the book was leading us to think more about Z for prime numbers. I was really excited to get into section 2.3 to discover what the properties are for congruence classes for primes since they were hinting at certain things so strongly. The most interesting part of the section was the proof of theorem of 2.8. First off, method one that they used to prove this theorem where they change Zn to Z is really clever and I had no idea that was possible. Then the arithmetic and the tricks used in the proof are really clever and helped me think more about tricks to use for proofs.

2.1, due on September 2

There were a few difficult parts for me to understand. During the proof of theorem 2.2, I got lost during the second step when they used -bc+bc=0. But later after rereading it, I realized that it was just a trick they were using and not something derived from an earlier equation. The first part of the proof of corollary 2.4 where the book says they have nothing to prove if the sets are disjoint didn't make sense to me. I don't understand why that is not applicable to the corollary. I didn't understand the second half of the proof for corollary 2.5 when they begin to suppose some inequalities. I didn't know where they got the s and t and how they came up with the inequalities and the rest of the proof built off of those things.

All in all I really enjoyed section 2.1. I enjoyed congruence and congruence classes when I learned about them in 290, because ever since I was in elementary school my dad would try to explain them to me and it was mostly nonsense to me until I took 290. I think the most interesting part of the section is the ending which states the corollary in which the set of all congruence classes modulo n has exactly n elements. I didn't come to that conclusion as I was reading the chapter but once it told me, it was such a logical completion of the section and makes a lot of the chapter solidified in my mind. I also liked the proof for theorem 2.3 because it reminded me that congruence can be proven by proving reflexive, symmetric, and transitive properties.

1.1-1.3, due on August 31

Honestly there were several difficult portions of the chapter. I didn't understand why the warning for Thm. 1.3 said that d is not necessarily the gcd of a and b since that's exactly what I thought the whole theorem was about. Further, I don't understand how d is the gcd by assumption but it is also the smallest positive integer. Isn't that contradictory? I also had a difficult time understanding the proof that followed. Then I was hung up on the proof of theorem 1.11 where they eliminate a prime on each side until they are left with the sums of the p's equaling 1. I have no idea where the 1 came from. There were a bunch of other places where I only understood most of the proofs so I will probably need to reread everything again.

The most interesting part of the material was the explanation of how Thm 1.1 The Division Algorithm is essentially showing how division is repeated subtraction. We talked about this idea a little in the one math education class we talked about, but it was very interesting to see how this is explained in general terms by proof rather than by examples of oranges or cookies as we did in the other class. I also finally understood prime numbers a little better and Thm 1.10 was really interesting and the proof made a lot of sense to me.

Introduction, due on August 31

 This is my fifth year at BYU. I need to finish submitting my paperwork to the Math Department in order to become an official Mathematics major.

I have take Calculus II (Math 113) and Calculus of Several Variables (Math 314), Elementary Linear Algebra (Math 313), Ordinary Differential Equations (Math 334), Fundamentals of Mathematics (Math 290), and Survey of Geometry (Math 362). However, it has been over a year since my last math class (Math 362) and as many as five years since Math 113.

I am taking this class because I need to graduate as quickly as possible and in a field that provides the best stepping stone for possible careers or graduate work. Because of my previous coursework, I have completed more of the Mathematics major than any other program. I do not know what career I want to pursue, so I am hoping to keep my options open as a math major. This is a required course and a prerequisite for other math classes required for graduation.

My professors for 290 and 362 were by far the most effective professors I have had in mathematics. Both professors would take many questions throughout the lectures and had a relaxed class atmosphere. My professor in 362 would sometimes allow some time at the end of class to work on homework with peers at our table. He would also allow a couple of minutes at the beginning of class to discuss issues we had with the homework with our peers so we could check our proofs and figure out if we had completed them correctly. If I remember correctly, the homework was based on completion which allowed me to be less overwhelmed with stress about getting exactly the right answer. My 290 professor was probably the clearest teacher I have had in explaining concepts because she didn't assume previous knowledge in her proofs. She also would explain and jumps she had in logic so the proof explanation was complete. Additionally, she provided us with ample study materials and review days for exams which helped me feel prepared and less anxious.

I am actually a transfer student now because I took a year off of school to get married and move to Kansas for my husband's job. We eventually decided it would be most economical and quickest for me to finish my degree at BYU instead of transferring to a school in Kansas. My husband is remaining in Kansas for work while I complete the rest of my degree. For that reason, I am hoping to graduate quickly.

The only office hour I might be able to attend is the TA hour from 4-5pm on Tuesday. I am available on Mondays, Wednesdays, and Fridays from 10-11am; Wednesdays and Fridays from 2-3pm; and Thursdays from 12-1pm or 3-4pm.