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.