9.4

1. I'm assuming this is not part of the curriculum, so the most difficult part is irrelevant...? I don't know. I saw this construction in Math 190, so it's all good.

2. It talked about the definition of addition, multiplication, etc. being "motivated" by the definition in Z and Q. I'm wondering a few things: which came first, the numbers themselves and their requirements, or some abstract notion of quantity and relationships...I mean, we could define things a lot of different ways, right? But 1/2 of a pie plus 1/4 of a pie is 3/4 of a pie. I guess the numbers are placeholders for some notion of quantity anyway, and that notion has to be consistent and maybe all of math is just what follows from that. I might be kind of an engineer at heart, but I keep asking myself what immediate applications I can find for this (and I mean this specifically). Is there any kind of tangible example of quotient fields outside of the rationals?

Exam 2

1. I think the First Theorem of Isomorphisms is important. I also think the nature of R/I when I is a prime ideal is important.

2. I think I need a firmer grasp of prime ideals and maximals.

3. Prove (log n)/n decreases monotonically.

6.3

1. This really wasn't that difficult of a section. I like the idea of maximals.

2. An integral domain is related to Z mod p where p is prime. Let ~ be congruence. If ab ~ 0 (mod n), with a, b both nonzero, then n must be composite since ab is a multiple of n (assuming n divides neither a nor b). Actually, typing this out, this isn't really a new concept. So I guess I'm making the connection now that I should have made before.

6.1

1. Ideals. It doesn't seem that bad. It's always harder to broaden my definition of things, but that's the nature of generalization. I mean, that's what we have to do, right? We can't conceive of everything and we can only conceive of things pretty close to our experience. So we generalize. We make sweeping claims from finitely many specifics. And proofs are generalizations too. But, uh, nothing difficult here. So I either really get it or I really don't.

2. I think it's interesting that even though we are broadening our definitions, congruence is still based on subtraction. I know that we could define that symbol to mean a lot of things (redefining addition and satisfying our axioms, of course), but it seems weird to me that we don't have a lot of different flavors of congruence (maybe we do...?) like difference congruence, logarithmic congruence, or cross-product congruence or something.

5.3

1. There wasn't too much difficult in this section. I am amazed at how similar irreducible polynomial quotient rings and Z mod p for some prime behave. Seriously, the fact that it is a field, and the implications of all that, well, it was unexpected to me.

2. I have this problem where I ask "why?" all the time. First it was "why are we learning this?" That was quickly mitigated as I renewed my faith in the BYU mathematics department--realized that they wouldn't teach things that one didn't need. It then shifted to, "Why was this developed?" Newton invented the Calculus (or discovered it, if you prefer) to model the laws of nature. I understand a little about the application of fields to cryptography and even to computation in general, but is the development of these ideas really that recent? (By recent, I mean early 1900s). Sometimes I have this suspicion that we are learning things that, while true, aren't immediately applicable, but they'll be assets when the fallout comes and we have to decide who gets to stay in the shelter and who doesn't. Kind of like that desert island game. I'm kidding. Kind of.

5.1

1. The most difficult thing for me was recognizing that the congruence classes for linear polynomials are unique for each unique l. p. and the corollary to that talked about what x^2+1 was related to.

2. Obviously, this has a lot to do with congruence classes for integers. I'm waiting for the big reveal when we see that all this information lets us send information securely via encryption or something very practical like that. Or maybe this is a course to prepare undergraduates for graduate work. What is a real world application of polynomial congruence classes?