Section 7.2, due on October 29

The most difficult part of this material for me was comprehending Theorem 7.8 and its proof. I think I understand it, it just takes some time to sink in. I also found it a little bit confusing just because there are so many different types of examples that it is sometimes hard for me to wrap my thoughts around exactly what is being implied.

The most interesting part of this section was Corollary 7.9 because it closely paralleled what we did with degree in fields and factors with integers. It is interesting to see similar concepts apply while dealing with things that can be totally different from each other.

Section 7.1 (the rest of it), due October 27

The most difficult part of this material for me was understanding the dihedral group of degree n. I don't really what is going on to make this a group. I also, found Theorem 7.4 a bit confusing with the different symbols. I think one of the most difficult things for me is that there are so many different "operations" that could take place that it is confusing to me to figure out exactly what a group is.

The most interesting part of this section for me was also the most difficult. I found it really rather neat how a group it seems is more or less something with a pattern and that pattern/operation can be a bunch of different things. I also gound it interesting that symmetry groups have been used by physicists to predict the existence of certain elementary particles.

Section 7.1-pg.164, due on October 25

The most difficult part of this material for me was understanding the symmetric group on n symbols. I am not sure how they came to all of these generalizations about Sn by just having looked at S6. I also found it confusing to try and see how they came up with the inverse of f under composition. I also don't understand how S3 had order 6???? I thought if elements repeated then they were just counted as 1 element within a set. Is this just a process of trial and error or is there some method to this? I was also wondering exactly what a group is, is it is "larger" than a ring, and if all groups are comparable to permutations.

The most interesting part of this material for me was the properties of S3 under composition. In another one of my math classes we are showing that if f and g are continuous, then so is their composition. I was thinking about this and trying to relate it to the example in this section. I am not sure how it relates, but it made me wonder if being continuous had anything to do with the fact that the composition operation is a group when dealing with integers.

Exam 2 review, due on October 20

1. The topics that believe will be important to know are roots and irreduciblity as well as some basics facts about a ring mod by an ideal, or a Field mod by an irreducible. I think knowing the idea behind Thrm. 4.5 will be important as well as Thrm 4.11. I think Thrm 6.1 as well as cosets, quotient rings and kernels will all be necessary to know.

2. I need to work on understanding quotient rings, cosets and ideals. I also need to clarify when certain definitions and theorems apply- in other words if you must have a field, a commutative ring, etc. I need to better understand different properties regarding degrees of polynomials. I also need to try and understand the First Isomorphic Thrm. better and understand when to apply it.

3. In the proof of Cor. 4.16 I don't understand why (c-a) can't be equal to the zero in F. The whole strategy of this proof is also confusing for me. In regards to cosets I was wondering if a coset of I in R means any elements of I plus any element of R where the elements in R are related to the representative? If this is true, does this mean that an ideal is always just some set of multiples of something? How does taking an element and adding something from I make it congruent to to something else in R mod I?

Section 6.3, due on October 18

The most difficult part of this material for me was connecting our past versions of prime with this new version and then using this new version in different proofs and things. It just takes me some time to adjust to the new use of prime and visualize it. It was also a little difficult for me to follow the proof of Thrm. 6.15.

The most interesting part of this material to me was the fact that R/P may not always be a field when P is prime. It's interesting to me how when you get more general, certain properties don't always apply. Yet this principle can also be true as you get more specific.

Section 6.2, due on October 15

The most difficult part of this material for me was tying to follow and understand what exactly is meant by the First Isomorphism Theorem and trying to understand all of its implications. I am not sure I fully grasp all that is going on with the homorphic image and the kernel and such and how it is all so closely related to ideals and quotient rings. I guess I don't really understand the structure of R/K.

The most interesting par of this material was the relationship all of this has to the things learned in Linear Algebra. Although I don't think I fully understand the homomorphic image concept, I did think it was really interesting when they compared it to photography. I found this an interesting comparison. It makes me wonder about how accurate the way we see things really is and how much of the exact structure we miss by simply observing things with our eyes.

Section 6.1 and 6.2 through the middle of pg. 147, due October 13

The most difficult part of this material for me was keeping all of the new terminology straight while trying to relate it to things we have done in the past so that I understand it. I don't really understand how the ideal generated by c1, c2, c3, .... is in fact an ideal. The cosets are also a little confusing to me, as well as the quotient ring R/I because I am confused as to how you "mod out" in this ring.

The most interesting part of this material for me was just the ideal itself. I thought it was interesting that there is a name for something that "absorbs products". Originally I would only think of the zero elements to be able to be ideals, but there is obviously more than that. The concept of ideals reminded me of the type of stuff you learn in genetics with dominant and recessive traits. To me an ideal was like this dominant trait that when combined with other stuff, still dominates (ie. when combined with other stuff it still ends up in the ideal).

Section 5.3, due on October 8

The most difficult part of this material for me was conceptually understanding the extension field and trying to visualize it. I am also confused when they say the F[x]/(p(x)) contains a root of p(x), if that root is a class or just a number. And if it's a class, is it the same type of class as the classes in F. Also, I was wondering if every elements in F is a congruence class, or not necessarily.

The most interesting part of this material for me was trying to think about why it would be useful to have the congruence-class ring. I also thought it was interesting to see how many levels the properties of integral domain, irreducibility, etc. can be carried out and still apply. It seems like you can always just keep finding bigger and bigger areas to work in, which kind of boggles my mind. I wonder if these applications have anything to do with different dimensions in space.

Section 5.2, due on October 6

The hardest part of this material for me is distinguishing between the field F, F[x], and the ring F[x]/(p(x)). I have to keep reminding myself that elements of F[x]/(p(x)) are classes of polynomials and not just polynomials. Theorem 5.7 was also confusing to me because of my confusion with trying to distinguish between F and F[x]/(p(x)).

The most interesting part of this material was Theorem 5.8. It was a little confusing to me, but it seemed like a cool fact. I was a little confused about the difference between Theorem 5.7 and Theorem 5.8, but when I figured out how they were different I thought this was also interesting. This section also makes me wonder what types of things the ring F[x]/(p(x)) is used for and why its structure is so much more rich then the set of congruence classes modulo n.

Section 5.1, due on October 4

The most difficult part of this material was trying to understand congruence classes from this perspective. Trying to figure out possible remainders is a little more involved in some F[x]. I would like to know if there is some trick I am not quite understanding that would make me think about this stuff in a faster way. It seemed practically the same yet trying to figure out the equivalence classes for a particular polynomial actually helped me to better understand congruence classes in general.

The most interesting part of this material was observing just how much everything related to the class of integers modulo n. Almost every proof was similar to a past theorem we had already learned. I also thought it was interesting that there are n^k distinct congruence classes in the set of all congruence classes modulo p(x). This is a little different from what we learned before, but still very similar.

Section 4.5-4.6, due on Oct. 1

Some of the hardest parts of this material for me was the section with complex numbers. I feel as though I am less familiar with imaginary properties in general, so it was hard for me to visualize what was going on. Also, I didn't quite understand the proof of Thrm. 4.22. I think I just need to familiarize myself with degrees of polynomials to better follow what they are saying.

The coolest parts of this material were the Eisenstein Criterion that is a way of showing f(x) is irreducible in Q(x). I also liked the Rational Root Test. I remember the first time I learned that concept and I thought it was cool, but had no idea where it was coming from. To see the proof of this test was really interesting for me.