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.)