6.2 (second half), due February 28

I'm having a little trouble because the new stuff about kernels and the connections with homomorphisms made since in class and I understood it then, but the things they have extended it into aren't making sense to me know. For example, with theorem 6.12 I have a hard time even understanding what it is saying although the logic behind the proof makes sense. I also don't think I'm seeing all of the implications of an injective homomorphism versus a surjective homomorphism. The explanation between Theorem 6.13 and it's proof seems like it's really important and a good connection to make, but I just can't seem to understand it. Another weird thing is that I understand the examples at the end of the section just not most of what leads up to them.

6.2 (through pg 147), due February 25

So I'm pretty confused these days, but it seems to me the first part of this reading is mostly just coming up with different terms/notation to describe things we have already discussed and learned about in a way that will help us to draw new connections...? This is kinda cool I guess because we have another way to make connections and new ways to talk about ideas that we've already learned. The most confusing thing for me is the third example because I didn't understand why they were adding ( 1 + I ) + (1 + I ). I'm also still a little confused by cosets and how it is that if I is the ideal of polynomials with even constant terms, then there are two distinct cosets. I think I'm not thinking about them in the right way.

6.1 (second half), due February 23

I've having a fairly hard time thinking about mod I where I is a subring because (like the book said) it's arbitrary and I can't think about it with any specificity. The whole coset thing seemed to just suddenly appear so that was also confusing. It's cool that we have this new way of looking at congruence and new ways of defining these things but I just don't understand it very well yet.

6.1 (through pg 138), due February 22

Things are really starting to feel abstract for me now. I like the new notation and way to think about congruence classes by defining a subring. After working with them for so long, this way seems intuitive, concise, and accurate. I did get a little confused toward the end of the reading where it talked about the ideal generate by c1, c2,... because r and c were both coming from R and I didn't see the point of that ideal.

5.3, due on February 18

The paragraph that talked about extension fields was particularly confusing to me. Partly because they use K to represent the field of congruence classes and then also the application where it says that although p(x) is irreducible in F[x], it may have roots in the extension field K. I haven't yet been able to sort out what that means or why it is true.
The section that discussed introducing new number systems and explained the complex numbers, how they work, why they're valid was definitely the most interesting part of this reading. I liked the little bit of math history thrown in there, and I also thought it was interesting that you can define complex numbers in this way and bring more meaning to them. It also helped to slightly clear up my confusion on the earlier part that I read, which is a little less fuzzy now.

5.2, due on February 16

The concepts in this section aren't especially difficult since we already learned them for congruent classes of integers, however it is a lot harder to think about congruence classes of polynomials. It's especially difficult for me to see how to reduce them (which classes are equivalent) and to manipulate them in order to show this.
I think theorem 5.8 is kind of cool just because you can apply it to any field and nonconstant polynomial where you have a commutative ring with identity. I think the main reason I think that it's cool is because we already know this theorem for integers and it's convenient that it transfers over.

5.1, due on February 14

This stuff is a little bit crazy. Equivalence classes with polynomials is a little much to wrap my mind around. I'm sure I'll get used to it and eventually think it's kind of cool, but for now it just seems like a pretty strange concept. At least the most theorems and proofs just cross straight over. 

Also, now our set of possible remainders just got so much bigger! Before it was so easy to check all the possibilities when using modulo and now the possibilities are endless. oh dear.

4.5, due on February 11

The most interesting part of this reading for me was to see why my teachers taught me to factor in the way that they did. Also it was cool to see why/how our method for finding all the possible roots of a function works.
The most difficult thing for me to understand was the proof showing that the function was irreducible, the math they did in the middle of the proof was the most confusing part and I didn't completely understand their justification for everything. I like Eisenstein's criterion for proving irreducibility much better (when it is applicable of course).

Due on February 9

I think the most important theorems are the ones that have specific names. and most important topics would be general definitions that are applied a lot such as what a field is, what a ring is, etc.

I expect computational problems as well as proofs

I need to work on knowing definitions and especially theorems better so that I have them in the forefront of my mind ready to use.

4.4, due on February 7

It's kind of crazy to me that  two different polynomials can induce the same function. That seems so contrary to everything I have learned before and it's a weird idea to think about. It's also kinda of cool though that you can just substitute all possible values of x in that ring, see that for each polynomial they map to the same elements, and then you can see that you really have the same function.
It's difficult to think of x has have two different roles. The hardest part of this is that it can change whether a statement is true or false based on what the meaning of x is in that particular context. Tricky!

4.3, due on February 4

I'm having a little bit of troubling distinguishing between units and associates. I realize they are connected but mean different things, the problem is that I can't tell in what ways they're related and what things make them different. It's just kind of a confusing definition for me.
This section was pretty short so there wasn't too much interesting stuff, but I do like that they transfer the basic concept of primes into the polynomials as well. It's also helpful that they describe it as "irreducible" instead of "prime" because it makes it easier to keep things straight and I think it describes the concept better anyway.