4.1

1. Once again I am burdened with the question: when are we just manipulating symbols and when are we actually manipulating ideas (obviously the latter is probably the answer and necessary for any rigorous study)? This chapter has challenged my conceptions of what x is. I guess even in my old way of thinking, it's just an element with certain properties. This actually did clear up my question of why we would say that polynomials in a ring had coefficients in that ring since it seemed stupid to have some polynomials in Z but use elements of R in place of x.

2. Talking about the degree of functions, I am wondering: how do you prove something that is obvious? The interesting thing is that obvious is subjective. I have been bored in class when a seemingly obvious concept has been proven. I have been confused in class when some method of proof is substituted with hand-waving because the result is obvious. Experience breeds intuition. We're all being taught to think in a certain way, to think with certain biases and conventions. But I think the world is too complex, and the subject matter far too advanced, for someone with real intuition to build it up from scratch.

Catch-up

1. I spend between one and two hours on each homework assignment. I spend probably an hour total reading the material and blogging about it, so between two and three hours total--nothing unreasonable. I feel the reading/lectures prepare me for the assignments.

2. I like the class. For now, as the material isn't too difficult, the class lecture seems redundant, but I can see the purpose of reading before, and I'm sure it will get harder, so it's all good.

3. It's a good class. It's well-taught. I think I can say that without pandering.

3.3

1. As the chapter said, this stuff isn't too hard with manageably-sized sets but as the sets grow, so does the problem of finding isomorphisms/homomorphisms by hand (or so it seems to me). This seems like a job well-suited for a computer: it is rigorously definable and really just symbol manipulation, which is what the Turing vision of a computer entailed.

2. Mathematics is for me often intriguing in that I wonder if a lot of math is developed (or discovered, but let's not start that debate...) but with no real purpose and is simply discarded. On the one hand, real world problems serve as motivation for mathematical (or any kind of) discovery that can lend itself to the solution of the problem. On the other, I could see many well-educated and well-read mathematicians with nothing really new to offer but who still develop some kind of consistent system and declare it new math. I wonder if Fermat knew the implications that prime numbers would have on our "futuristic" society (where almost all security is based thereon, etc.).

3.2

1. The most difficult thing for me in this section is keeping symbol manipulation separate from the logical validity of the operations. It seems interesting to me that behaviors of systems can be dictated by defined symbols and operations. I probably don't understand it, though, to be saying that. I think it's also interesting that kids are taught symbol manipulation through school and teachers struggle to teach how it is applied--there's just too much to go over and most kids just aren't willing to put forth the effort into something if it doesn't come easy. As a math tutor, I see this consistently and is the rule, not the exception.

2. I am taking Theory of Analysis (315) right now and the book started with fields. The author, who doesn't mince words, went through a similar process as this book to establish laws for fields, groups, etc., albeit with way fewer words. It's nice to see some mathematical convergence since for so long it seems there is just new topic after new topic.

2.3

1. I just had Math 190, so modular arithmetic is fresh in my mind. I know you probably hate that response, but I suppose I don't know what I don't know and so I can't think about it at a deeper level "at will". This presupposes that you are using this to determine what to emphasize during class time. If you would like to know the hardest aspect anyway, I suppose it would be...I really don't have anything at this point. Stay tuned.

2. One of my math professors said, "Mathematics is an opportunity to be honest with yourself." Like right now, I don't know the ramifications of a prime modulus producing a finite field but I have to assume that these concepts are taught for a) some application which I cannot come up with, or b) as a foundation to more advanced and applicable concepts. The great thing is that in the near future, I will have another tool at my disposal to solve problems. That's cool.

2.2

1. The most difficult thing I can foresee is proving that Z modulo n is only a field if n is a prime number. Actually, that won't be that bad, since the greatest common divisor of the prime number and all its congruence classes is 1 (except for [0]). I'm taking Theory of Analysis (315) and I have to prove, with fixed b > 0, that r = m/n = p/q => (b^m)^(1/n) = (b^p)^(1/q). Any suggestions?

2. I was talking to a BYU mathematics Master's student and he told me he could find no application for the infinity of infinities proof (that there are uncountable sets of uncountable sets). It got me thinking about all of the development of prime numbers and number theory in a time when it was unreasonable to use them for, say, cryptography. I can't see a use for finite fields yet, but I'm sure when I realize what I can do with them, I'll wonder what I ever did without them.

1.1 - 1.3

1. I would say the most difficult thing for me is accepting that negative numbers are primes. It flies in the face of most of my long held beliefs. But I suppose that even positive primes are divisible by -1, invalidating the definition that primes are only divisible by 1 and themselves.

2. I think that the fundamental theorem of arithmetic may have some ramifications for the methods of cryptography in use now (the RSA scheme). If I recall correctly, the security of the RSA scheme hinges on the fact that it is hard to factor large numbers. That's it, though. Computational power is still increasing at a fairly rapid rate (especially considering the use of FPGAs in GPUs) and who's to say an algorithm won't be developed to factor the numbers in polynomial time? With more investigation, I might be able to determine if the existence of multiple prime factorizations of a number compromised the RSA scheme, but at cursory glance it looks like it wouldn't (because the totient function would be different and so the keys would be different), but it did get me thinking.

Introduction

My name is Kimball Germane. I am a Junior with a declared major of Computer Engineering, but I am changing it to Mathematics with a minor in Computer Science.

Math classes taken since calculus:

Math 334

Math 348 (I think it's 348, the class was PDE)

CS 236 (Discrete Math)

Math 214

Math 190

Math 343

I am taking this class 60% to fulfill requirements for the major, 40% because I like math. But maybe I'm not so sure on what abstract algebra really is. I really liked Math 190. I wonder if it's anything like that...

The most effective math professor I have had is either Vianey Villamizar or Sum Chow. I had Dr. Villamizar for both flavors of differential equations and he can explain (what I think are) complex topics very clearly. I had Dr. Chow for linear algebra, and even though there was a little bit of a language barrier, his method of teaching was enlightening.

I've trained in Shaolin kung fu for over eight years.