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.