6.2 Part 2

1. What was the hardest part of the reading for you?

Maybe it's because the first part of this reading was already covered in class, so that stuff was pretty easy, but I struggled with the First Isomorphism Theorem. I read through it a couple of times, and I still don't really understand exactly what it's saying. Usually when that happens I can understand it by reading the proof, but the proof in the book mostly confused me. It just has so many words!

2. A Rant:
 It's silly, and a little embarrassing, but  I wanted to scream when I realized they were using "pi" as a function, rather than functioning as the "pi" we all know and love. Why couldn't they use any of the other 24 letters of the alphabet (not counting the commonly used f and g) or any of the other Greek letters at their disposal?!? It's silly, I know, but that really made me stop and think each time they used it. WTF.

6.2

1. What was the most difficult part of the reading for you?
I think the only thing that is becoming a little annoying/difficult for me by now is the stupid notation. Everything else so far seems pretty nice and easy as it comes from everything else we've been talking about. Though, it did make me laugh and annoyed when they talked about the three different uses of "+" on page 146. Mostly I just rolled my eyes.

2. Say something else amazing.
Um.... I mostly don't know what to say since this reading was so quick. I hated the homework for the second part of 6.1. I thought I understood everything, but doing the homework kind of confused me and annoyed me. The end.

6.1 Part 2

1. What was the most difficult part of the reading for you?
I'll be very specific. I think I'm savvy on most of this reading, but the basic part where I think they were summing some important stuff up on page 140 (at the top) kind of lost me. Maybe it's the notation, or... I don't know. But this "coset" stuff is a bit confusing to me. It makes sense that there would be congruence classes, but.... I don't know. I guess that's just how I feel. It's late.

2. Why is 2+I=[2] a congruence class in the principal ideal 3?
Alright, so 3=0(mod I) since 3=3k in I, and [3]={...-6,-3,0,3,6,...). So, 2=5(modI) and -3=2-5=3k.... I think this makes sense. Wait, does it make sense? I think so, but.... oh whatever. What a pathetic blog.

6.1

1. What was the most difficult part of the reading for you?
I think that I can follow the concept of ideals pretty well. It seems to make sense to me, and that is good. The thing that I'm struggling with, and it could be my level of focus, is the distinction between the different ideals that the book talks about.

2. Write the definitions of the different ideals:
IDEAL: A subring I of a ring R is an ideal provided that whenever r in R and a in I, then ra in I and ar in I. Note, a nonempty subset I of a ring R is an ieal if and only if it has these properties: i) if a,b in I, then a-b in I; ii) if r in R and a in I, then ra in I and ar in I.

LEFT IDEAL: Like the fourth example on page 136 -- like a matrix that seems to have an ideal for ra but maybe not for ar.

PRINCIPAL IDEAL GENERATED BY C: Let R be a commutative ring with identity, c in R and I the set of all multiples of c in R, that is, I={rc | r in R}. Then I is and ideal. (yeah, I still don't really get this)

IDEALS GENERATED BY C1, C2,...., CN: Let R be a commutative ring with identity and c1, c2,....,cn in R. Then the set I={r1c1+r2c2+....+rncn|r1, r2,....,rn in R} is an ideal in R. (again.... I don't really see it.)

5.3

1. What was the most difficult part of the reading for you?
I am kind of confused by the idea of the "extension field". If I just blindly accept it, and follow the reading, I understand it's significance and can see why it's important to building complex numbers, but.... the whole a=[x] and plugging that in for each class kind of blew my mind a little. Mostly, watching x^2+x+1 becoming [x]^2+[x]+1 then equaling zero was kind of... crazy.

2. Where I stand
Honestly, I struggled with the 5.1 homework a bunch.  I am really not struggling with any of the correlation between Zn and these [f(x)] classes and such, but I struggle to remember all of the theorems and corollaries for everything. Mostly they confuse me. Also, I am kind of having a hard time accepting the fact that [x+1] isn't the same as [2x+2] in Z[x] or Q[x] or whatever. Does that make sense? Why aren't monics important here? 

5.2

1. What was the most difficult part of the reading for you?
I think the author knew that this would be confusing, because he kept talking about how it was difficult to grasp, but I don't really understand the significance (or even the idea) behind of having the subsets within the  rings.... or whatever that was. The stuff explained in the example on page 126. It made me laugh because he kept saying, "You know, just like that one example on page 126" like that would make the world better or whatever. I still don't really get it.

2. So.....what does it mean? That example that is...
The best that I can imagine is that 0,1  (as opposed to [0],[1])  is closed under addition and multiplication and probably subtraction, so I guess that is why that is a subset. But.... who cares? I don't get it. I'm just not sure if it matters that they only be 0 and 1 or if the subset may also contain another non-0 and non-1 element.... I just don't know. I think I just don't get it because I'm not sure why it even matters. But hopefully we'll see why later.

5.1

(sorry, I'm feeling pretty sick as I'm writing this, so it'll be short and sweet)
1. What was the most difficult part of the reading for you?
Though there were examples given, perhaps it's because of how I'm feeling, but I'm still not exactly sure what this is saying about polynomial congruence. Well, I guess that isn't true, but the first example of page 122 kind of scared me. I'm going to reread it when my head isn't hurting my soul, but for now I'm just intimidated and scared.


2. Helping to understand
The thing that I think will help me the best is similar to something that I said way back in chapter 1, and that is to always write stuff out when it comes to congruence classes with polynomials the exact same way that I did for the integers.

Exam 1 Prep

1. Which topics and theorems do you think are the most important  out of those we have studied?
If I had to guess, I would guess that knowing the different axioms and how they define the different rings seem like they would be the most important thing to know, since it is the foundation of all of the important/non-reviewey stuff we've done. It seems like if I can nail those axioms down, and use them as definitions to help with different theorems and such will be very helpful.

2. What kinds of questions do you expect to see on the exam?
I am kind of imagine that the types of questions we'll see on the exam will be pretty similar to homework questions and questions we've encountered in class.

3. What do you need to work on understanding better before the exam?
I really need to study the 3 proofs from the study guide. I'm mostly nervous about the wording and what past info can/can't be used for these proofs. Similarly, I'm a little nervous about congruence between rings. The idea never worried me, but being able to see the congruence does make me a little worried.

4.4

1. What was the most difficult part of the reading for you?
Well, I'm not going to lie, I giggled when the author of the book seemed to be speaking right to me. The thing that is worrying me the most about this reading is being able o tell when a polynomial is a polynomial in F[x] or when it is the 'rule of it's induced function' as the book says. I'm sure that context will have to play a big role in this, but I just feel programed to treat polynomials as functions, not as...objects... But, as the book says on page 101: "the practice of using x for both is so widespread you may as well get used to it." So, this is me, getting used to it.

2. The remainder theorem
I have to admit that my mind was blown a little bit by this theorem. I was so shocked that the remainder could be determined so easily and as I read the theorem I thought to myself, "How is this possible?!? I want proof!" And the proof was so simple and easy to follow, I just had to laugh a little and say, "How did you not get that before?" Either way, it seems fun.

4.3

1. What was the most difficult part of the reading for you?
Oh my gosh, maybe it was because we had to do homework right after doing some reading for this section with out a class period to digest, but I feel pretty lost. I'm not confused by the idea that something is "irreducible"  or "reducible" but trying to do the homework tonight proved to me that I don't know how to tell for sure when something is irreducible or not. Also, this "associate" thing.... what is that? I don't understand it. I mean, I guess I understand that it is the product of a unit and something else, but I really don't understand much else about it, and that makes me sad.

2. Why I felt lost on the homework
So, I'm specifically talking about 4.3 2 and 9. I really struggled with 2 because I really don't understand associates. I understand being non-zero, I understand monic, and I understand unique. That being said, I don't even understand what I'm being asked to prove in number 2.
For number 9, wow.... Is there an easier way to do this? What I did for parts a) and b) is I wrote down every possible function I could imagine (for part a I wrote down a list of the polynomials with degree 2 in Z(2)) which weren't that many. Then, looking at them, I just eliminated things I knew could be factored. Then, when I remembered I was in Z(2) and should be able to eliminate some more things, I wrote down the two possible factors (x and x+1) and wrote the 3 possible combinations of the two. (x^2, x(x+1), and (x+1)^2) and eliminated those options too, leaving me with the one irreducible thing. Well, I was able to do that for part b, but it took a little more time. Then, realizing that I had to pretty much check 27 different combinations for part c made me tired and I just feel like there must be some easier way to see if things were irreducible or not. I just don't know what it is. Seriously, what am I not getting here.

4.2

1. What was the most difficult part of the reading for you?
Tonight, I got nervous for doing the homework for this section. Nothing really stood out to me as particularly difficult, but realizing that we're probably going to have to use the division algorithm a couple of times to find the gcd for two polynomials worried me. I have a hard enough time dividing polynomials cleanly on my paper so that I can follow it, so having to do it several times makes me a tad nervous, but I guess we just take it one problem at a time.

2. A caution for the future:
Something that worries me, mostly because I know how forgetful I am, will be to convert polynomials to "monics" when calling them the gcd. Hopefully by talking about it in this tiny paragraph will remind me to make sure that the leading coefficient is 1 when talking about it.... I think I can.... But man, I bet finding the formulas (like, the reverse of the division algorithm) is going to suck....