Test 1 Review Due September 29, 2011

1. Which topics and theorems do you think are the most important out of those we have studied?
It seems to me that The Fundamental Theorem of Arithmetic will be important (I mean, it does have the word "Fundamental" in it. Also the Phi Function, Chinese Remainder Theorem, and Quadratic Reciprocity all seem to be big ideas that were built upon other smaller ideas we've discussed.

2. What kinds of questions do you expect to see on the exam?
I expect us to have to solve a system of congruences (using the chinese remainder theorem) as well as finding if a number is a quadratic residue mod whatever using that mess. Also, I'm guessing there will be some questions about rings and probably a proof or two about primes.

3. What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Friday.
As is pretty obvious, my frustration over the homework that was due Monday seemed to put a damper on what I thought had been pretty good progress. I've pretty much given up for this week (with every intention to pick myself up at the start of next week...) But, I am still really confused about the whole Quadratic Reciprocity thing. I thought I understood how to flip and solve and yadda yadda at the end of class, but then I started to do the homework and got confused and frustrated. I'm mostly confused because from the theorems in that chapter (and with the jacobi stuff) it seemed that your numbers had to be, at the very least, odd. Then you were saying something about using the chinese therorem to break things up and whatever.... I revisited the chinese theorem, and I was still pretty confused. So, I need to understand that stuff better, and I'd like to see an example done, like the one in the homework, where the numbers weren't necessarily prime, and one of them was even -- so you had to use the CRT. After that it'll just be up to me to decide how much I'll be watching conference, and how much I'll be studying... sigh... what a spiritual dilemma....

Section 3.1.1 Due on September 28, 2011

1. What was the most difficult part of the reading for you?
I.... hmm... I am kind of confused by some of the different proofs that explain the infinite number of primes. I'm aware (it makes sense to me) that a prime cannot divide p1p2p3...pk+1, but I'm not sure how that is proven. I know that that isn't the most important part of those proofs, but it is something that has me caught up and distracts me. Aside from that, I struggled to follow the factorial proofs... I guess that's all I have to say about that.

2. What was the most interesting part of the reading for you?
I thought it was interesting that you could prove something about the number of primes using polynomials. I didn't follow the proof very well, but it seems interesting -- and complicated. On the other hand, the proof that made the most sense to me was the one that used the Euler phi function.

Section 2.6 Due on September 23, 2011

1. What was the most difficult part of the reading for you?
Oh gub. I don't want to sound like a moron. I'm serious when I say that I read these sections 2 or 3 times before blogging about them (always after class, usually that night, then the next night before class). I'm not sure why things don't really make sense to me while reading them... but this section really really confused me again. (The last section really confused me too, but after class it made more sense to me, so hopefully the same will be said for this one too.) But here, my problem begins at the roots of the section. The lemma (2.6.1) really doesn't make sense to me. Not even after reading the proof. There's just to many if/then/if and only if's for me to follow the logic of what is being said. Unfortunately, it goes down from there.

2. What was the most interesting part of the reading for you?
Well, I guess I'm going to have to take a hit on this because I didn't think anything was interesting because I didn't understand anything in it... Oh well. Cest La Vie! I hope you are having the most amazing trip of your life.

Section 2.5.2 Due on September 21, 2011

1. What was the most difficult part of the reading for you?
Okay... I've now read this section 3 times (all on different parts of the day) and I can honestly say that I struggled through most of it, which is embarrassing because it doesn't seem like it should be too difficult. I guess that most of my confusion is coming from when to do what or whatever. I'm kind of confused by how many solutions you get when (mostly Theorem 2.5.2.3 and really that whole page kind of confused me.) Also, this quadratic residue and quadratic nonresidue malarky didn't make much sense to me.

2. What was the most interesting part of the reading for you?
I thought that the fact that you can still use (sometimes) the quadratic formula was pretty interesting. But... when can I use it? The theorem says when p is an odd prime (so, not 2? Isn't that when we normally use the quadratic formula?).  In the second line of page 45, is there a typo? I mean, what's an inverse of Z7? What does that even mean?

Section 2.5.1 Due on September 19, 2011

1. What was the most difficult part of the reading for you?
Well, they say that Chinese is one of the hardest languages to learn... I've gotta say that if the language is anything like the Chinese remainder theorem, I'd agree. I'm not exactly sure what it's trying to say and I certainly couldn't follow the proof. As best as I could tell, it was like solving a set of equations where, if we were in the integers, you'd line up your equations and subtract, or solve for one x and plug it into another, etc. But, I must say that I was pretty lost when it came to the example they provided. It just looked like a mess to me.

2. What was the most interesting part of the reading for you?
I guess that for me, the most interesting part of the reading was solving for x mod Zn when n wasn't prime and (a,m)=d (as described in the book). It made sense to me that there would be more than one solution . I guess that I would have just written out a table or something to solve these problems, but obviously that isn't really rational all of the time. But their method was helpful in reminding me of the nature of Zn when n is composite.

Section 2.4.5 Due on September 16, 2011

1. What was the most difficult part of the material for you?
First of all, I'm glad that that example in 2.4.4  really was wrong, because I was ready to throw up. I also found this section easier to follow than some of the others. I think that part of that is because a lot of the things discussed are almost "restating" things that we've already covered. However, I did kind of struggle with Theorem 2.4.5.1. If I were to state it in my own words, I might say "If G is a finite cyclic group of order n, any number that divides n, there exists subgroups of that order." So, I'm pretty sure that is what the theory is saying, but following the proof (since I always struggle with contradiction proofs) kind of lost me.

2. What was the most interesting part of the material?
I really wish that we could make a big chart of the phi function's functionalities. Like, numbers that divide n (in Zn, or whatever) can tell you the different sizes of subgroups, or a number d such that (d,n)=1 will be a generator of that group, or something... It just seems like there are so many similar things for things dealing with the phi function that I know I should probably make a chart or list them out or something so I can compare and contrast all of these theorems that are bleeding together for me.

Section 2.4.4 Due on September 14, 2011

1. What was the most difficult part of the material for you?
Wow. No joke, I'm struggling with a couple of things in this chapter. As I was reading it, and things were getting jumbled together, I kept thinking "Geez Loueez, I wish they'd just give an example or two." Then, they gave two examples. I definitely followed the first one, but the second, holy cow. Is it just me? or are there some issues with 2.4.4.2? I mean, in that table it has a 1 beneath the x. Shouldn't that be an x(bar)? Also, why does the number 1 have order 4? Shouldn't 1 have order 1? And 2 has order 2? Doesn't 2x2=4? Not 8 or 15? (I wasn't sure which it should equal in this case. In fact, I'm kind of confused about this group of units from a non prime n for mod n. Does that make sense? I really remember (and almost enjoy) cyclic groups, and I remember orders of groups and elements, but this chapter lost me.

2. What was the most interesting part of the material?
Hmmmm... this is a tough on because I really struggled with this chapter. The entire time I was reading it I was kind of wondering if we'd be talking about that thing we did in Abstract Algebra last year. I don't remember exactly what it was.... but with these cyclic groups we divided them into subgroups where those subgroups were all of equal order and I think it had something to do with associates. Crap. What was that called? I don't even remember. Anyway, I just kept thinking about that as I was reading page 31 that listed so many theorems.

Section 2.4.3

1. What was the most difficult part of the material for you?
I'm confused about a couple of things. First of all, I'm not exactly sure what a "Reduced residue system" is... I've read the definition several times, and I still haven't quite figured it out. I guess I'm confused by the last line in that definition.
The other thing that I don't quite understand is that big pi/n looking thing that first shows up on theorem 2.4.3.1.

2. What was the most interesting part of the material?
I think that we talked about this in abstract algebra, but I feel like we used a different name, or notation, or formula, or something. Maybe I should double check my notes. But, I find the Euler phi function pretty interesting. I'm not exactly sure why it's useful yet, but I'm sure we'll find out before too long since the name of this book is "Number Theory: an Introduction via the distribution of primes."

Section 2.4.2

1. What was the most difficult part of the material for you?
It's a small paragraph, and maybe I just need a review on quotient rings and such, but the paragraph that starts "This theorem is actually a special case..." on page 24 kind of lost me. Maybe it isn't so important right now, since I followed everything else in this section, but that was the most difficult part of the reading for me.

2. What was the most interesting part of the material?
Initially I was going to say that Wilson's Theorem (2.4.2.3, pg. 26) was the most difficult part of the reading. But, as I was re-reading it in preparation for this blog, one sentence at a time, I thought the logic was really fascinating. I'm glad it wasn't assigned as a homework problem, since I don't think I would have ever thought about proving it in that way, but it really helped me revisit modular arithmetic in a way I hadn't thought of yet this semester.

Pg. 21-2.4.1

1. What was the most difficult part of the material for you?



I have a pretty good grasp on associates and units and such (I think.) Like, I understand that in Integers, 6, -6 are associates because 1,-1 are units.... But I am struggling with the UFD (Unique Factorization Domain) So, let me get this straight, if D is an integral domain, then if ab=0 either a=0 or b=0. Okay, so for d in D, either d=0, d is a unit (1,-1 for integers), or d has a factorization into primes (okay, that's that fundamental theorem of whatever.) But what I don't understand are the theorems that follow. Like, the Euclidean domain, or Gaussian integers. So, for the Euclidean domain, it says that if there is some function N: D\{0} to N+{0}, (or, a function that maps, in the example of integers I'm using, all integers except for 0, to the natural numbers plus zero) For each a,b in D where a=0 (would b be zero? I mean, it seems to me like the domain can't be zero since it's D\{0}), there exist q,r also in D such that b=aq+r and either r=0 or r doesn't equal zero, and N(r)<N(a). So, I'm confused as to when there is and isn't a zero here. I'm also confused at what it's saying at the end: N(r)<N(a). Also, if I'm understanding this correctly, it just sounds like the euclidean algorithm to me. Like, if we start with some d in D, we could take the absolute value of it, and that's our function mapping it to N. We could even say that D(0)=N(0)=0. So, then is the N(r)<N(a) just saying 0<=r<a?
I guess I'm mostly confused at what this is saying, and if it's any different than the division algorithm.


2. What was the most interesting part of the material? 
This reading was pretty short, so I'm not sure what was "interesting" to me. Maybe I can get away with saying that I don't remember the term "Residue" being used in 371. Is a complete residue system just the same as saying all of the classes that are congruent to one another? Like, in modulo 3, [0], [3], [6], [9].... all of those is a complete residue system? No, wait, I just re-read it, and it seems like a "complete residue system" for mod 3 would be [0], [1], and [2]. Right? That's how I understand it. Isn't changing books just so annoying? It's like... such a change of vocabulary and definitions, and whatever. (I know it's not your fault, and I'm glad I didn't have to buy a book, I just wish all authors used the same brain.)

Section 2.3

1. What was the most difficult part of the material for you?
Oh man. I knew this day would come, but I didn't expect it so soon. I struggled with this in Abstract algebra, and here we are again... And I'm still struggling... Hopefully after class tomorrow, I'll understand it better... But enough of the "..." What I'm really struggling are the two proofs about how there is an infinite number of primes. I understand breaking numbers down into primes, and how they are unique, but the fact that they are infinite still hurts my head.

2. What was the most interesting part of the material? 
Though I thought 2.3.4 was kind of confusingly stated, it was through trying to understand what it was saying that I actually understood this theorem better. I remember we talked about this last semester too, and I kind of understood it, but I know that I get it better now. Also, I really like this way of finding GCD and LCM. At least, I like it much more than the Euclidean Algorithm.