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.
Sunday, February 27, 2011
Thursday, February 24, 2011
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.
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.
Tuesday, February 22, 2011
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.
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.
Monday, February 21, 2011
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.)
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.)
Thursday, February 17, 2011
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?
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?
Tuesday, February 15, 2011
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.
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.
Sunday, February 13, 2011
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.
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.
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