- Difficult: I understand how to find a homomorphism and all that, but what I don't understand is why? Why would it be important to find an ideal homomorphism? I guess I probably shouldn't ask the why question, because I feel that way about almost everything in this class, but that's the main thing I was wondering while reading this chapter. Ideals are important because they generalize congruence, but what benefit comes from being a homomorphism?
- Reflective: This section seemed to be the one that varied the most. In the past few sections I've easily been able to recognize parallels to the previous chapters, but not so much in this one. Did I miss it? Or are we staring to change topics?
- Difficult: This chapter actually seemed pretty straightforward. Once I was able to wrap my head around the whole a+I thing, the rest was easy. However, I'm still a little confused on what R/I means. Is that just another notation for a+I? Why do we need the two notations?
- Reflective: This is pretty much exactly the same as what we learned before, which is nice. So long as I don't allow myself to get hung up on the new notation (which is actually really hard to do) then this doesn't seem so bad.
- Difficult: The most difficult thing for me in this section was understanding exactly what I is. Because I is a subring. And so wehn they have things like a is congruent to c (mod I) iff a+I=c+I it doesn't make any sense to me. How can you have a+I? What does that look like? I just can't visualize it. Also, I wasn't sure what a coset was.
- Reflective: I like that the relation of congruenec modulo I is an equivalence relation. Things always seem to work out nicely when that happens. I also liked the example at the top of pg. 139, that was very easy to understand (unlike most of the examples in this book).
- Difficult: This chapter actually made a surprising amount of sense. I guess the most difficult thing for me is actually visualizing an ideal. Because even though I could tell you what one should be like, I don't know if I could actually recognize or create one. I look forward to examples in class.
- Reflective: Theorem 6.1 states that if a,b in I, then a-b in I. But if both a,b are in I, then that means that a,b, have a common factor, right? Just like in the first example where it was mod 3. If a,b in I, then a-b=3(c-d)=3(e) so of course c-d 3e would be in I, so a-b is in I. I'm I thinking of this correctly?
- Difficult: In Theorem 5.10 it says that p(x) is irreducible in F[x]. But why is that necessary? A field didn't JUST have prime numbers, so why does this one have just irreducible polynomials? It doesn't make sense to me. Also, I have no idea what an extension field is. I've read that part a bajilion times, and it's still jibberish. Looking forward to the explanation in class.
- Reflective: Hm... For once, it's actually the reflective part that's hard. "What was the most interesting part of the material?" Um... well... I didn't understand what an extension field was, and that's pretty much what it was all about. So... "How does this material connect to something else you have learned in mathematics?" well, so far as I can tell it doesn't! "How is this material useful/relevant to your intellectual or career interests?" I'm not sure. I wish someone would tell me how AA is going to help me teach high school math.
- Difficult: In theorem 5.7 it ends by saying "Furthermore, F[x]/(p(x)) contains a subring F* that is isomorphic to F." This doesn't make sense to me. It's just so vague. Why does it contain a subring? What's in this subring? Why does that subring have to be isomorphic? This theorem just doesn't make sense.
- Reflective: I feel like we're learning the same thing over and over again just with little twists. You'd think that that would make things easier. And I guess in some degree it does. But I do kind of feel that there's a lot of "duh" stuff that the book explains, but then it doesn't explain some of the harder stuff. I know this seems like it should be more under the "difficult" bullet, but this is what I was "reflecting" on while reading this section.
- Difficult: Woah. There was a whole lot of weird punctuation going on in this section. But I guess it's about time we addressed division in rings/fields. I think the most confusing part for me was the idea of dividing equivalence classes. Because you can't have an equivalence class of 3/5, can you? Equivalence classes have to be whole numbers. So that doesn't make sense.
- Reflective: The book writes it as (a,b)~(c,d), and at first this really confused me because it then said ac=bd which looks like it's doing some kind of weird F.O.I.L. thing. But then I realized that it's just the first number in the parenthesis, divded by the second number. So a/b=c/d. This was easier for me to understand than ac=bd, and once I got that, the chapter read much easier.
Which topics and theorems do you think are the most important out of those we have studied?
- I think it's most important to know the definitions of all the different terms for the test. If you know exactly what all the terms mean, then you can more fully use that knowledge on the exam. Also, theorems with names are usually pretty important.
What kinds of questions do you expect to see on the exam?
- I honestly have no idea. This class has been an odd mix of both computational problems and proofs. The sample questions have a lot of "prove or disprove." So... probably a lot of that. But I HOPE they are computational.
What do you need to work on understanding better before the exam?
- I really have no idea how to do proofs. I just lack that intuition to know where to start or what to do. I really hope that by learning the definitions better I'll know what I need to do on a proof. Because right now I am scared to death.
- Difficult: I understood that if something is irreductable that means that it is basically a "prime" polynomial, but some things in the definition confused me. What does it mean that its divisors are its "associates"? I read the definition of associate, but I didn't get it. Also, even though something is irreductable, can it still be divided by a constant term? Because the definition said "its only divisors are its associates and the nonzero constant polynomials." But that doesn't sound very prime-ish to me.
- Reflective: I love how closely related these polynomials are to the "normal" rings we've been studying. I've found it very interesting to extend the idea of a ring to a new form. And I'm glad things haven't changed too much because I really feel like I'm building up my knowledge rather than having a bunch of new concept trying to take up space in my brain.
