7.1, due on September 30

Difficult:
    I hardly follow this at all. I don't understand why they can resolve 2 dimensional coordinates into a real field extension, and that's probably the smallest of my issues, as I do not have enough understanding to formulate a sensible question on what remains. A second look makes it slightly clearer what's going on, but it is still bizarre to me.

Reflective:
    Using algebra to solve geometry is not a new idea, as the Cartesian coordinate system demonstrates, but using abstract algebra to solve the question of what is constructable in a traditional Euclidean manner is bafflingly brilliant.

Progress:
    As far as preparations go, the lectures are helping far more than the book. I only read ahead because it is an assignment to do so, although I did find that I was more lost than usual on the day that I forgot to do the readings, so I suppose that they are helping prime my mind to learn what is actually going on.
    Homework has not been too much of an issue for me to understand. The lectures have adequately prepared me for it, and I can't think of how the format could be improved.

6.2, due on September 27

Difficult:
    It took me a while to follow example 6.5. They better explained the similar example 6.8.


Reflective:
    Example 6.8 used a neat trick to show that [Q(sqrt(2),sqrt(3),sqrt(5)):Q(sqrt(2),sqrt(3))] = 2. It took me long enough just to accept that pq+3rs = 0 and the other similar equations. Noticing a relationship between possible solutions is just not a trick for my mind.

5.4 & 6.1, due on September 25

Difficult:
    I was massively confused on reading this material. The ideas still make sense, but the explanations and notation are all going over my head. For instance, I assume the square bracket notation at the end of lemma 5.14 is about dimension, but I can't remember for sure. Also, the notation in definition 5.15 was confusing until I realized it was about commutativity. I even had to look up what a basis was to understand lemma 5.14. This is why I need a professor to explain math to me.

Reflective:
    I do like that the commutativity idea from definition 4.12 finally got utilized in theorem 5.16. Likewise, it pleases me to go back to the vector spaces of linear algebra, though the notation is all lost on me now. I feel like Shakespeare in this respect, of whom it was said that he could tell you an excellent story as long as you told it to him first. For all the intelligence I claim to have, I can only seem to learn some things from other people.

4.2-4.3 & 5.1, due on September 20

Difficult:
    The material is rather straightforward. I suppose that my only difficulty at this point is that I don't know where this is going. Transcendental numbers are cool, but what can you do with them, or why should I care about simple extensions?

Reflective:
    The concept of transcendental numbers is cool. Their existence isn't totally expected until you realize that there are only countably many polynomials of finite order over the rationals, as aleph-null squared is still aleph-null.
    I also didn't quite expect that simple extensions could be a composition of simple extensions, for example adjoining Q with i followed by sqrt(2) is the same as adjoining Q by i + sqrt(2).

4.1, due on September 18

Difficult:
    Like last time, I found the material too intuitive to really follow the formalism of the proofs. The danger in this is made apparent by example 4.9.3, where Q is adjoined by the cube root of 3. Without such a formalism to use, patterns found to work for other subfields would be assumed to work for any subfield without proof, which is show to be false in this example. 

Reflective:
    It is interesting that the definition for a field extension used the language of monomorphisms rather than superfields. It mentions that this is for technical reasons, but I would be most interested to see what they are. I would assume that at some point we step away from using a subfield of C to define field extensions at some point, and use formal roots that are assumed to exist instead, like was done in Math 371.

3.5-3.6, due on September 16

Difficult:
    The material was so intuitive that it was hard to follow the formalisms of the proofs. It is also unintuitive why Lemma 3.25 was proven, while reducibility holding over the obvious homomorphism from Z[t] to Zn[t] was left unproven. Along those lines, the general sloppiness of the proofs in this book are hurting the formal parts of my brain, as I tend to be similarly sloppy with my first draft proofs, so I'm not getting as good of a feel for the formalisms of the material.

Reflective:
    The fact that irreducibility in Zn[t] implies irreducibility in Z[t], is quite a useful fact. if f in Zn[t] is irreducible, then any g in Z[t] that maps to f in Zn[t] is irreducible, and the number of g's that fit that criterion for a given f are infinite. Countable, assuming degree(g) < infinity, but still infinite. Also, I noticed that by that criterion, a similar criterion exists to the Eisenstein criterion. If f(t) in Z[t] = sigma(an*f^n) and m|an for all n except m does not divide a0, then f maps to g in Zm[t] where g = a0, which is irreducible.

3.2-3.4, due on September 13

Difficult:

    The "obvious" fact that f(t) is irreducible iff f(t+1) is irreducible is not obvious to me at all.

Reflective:
    I think it is interesting that I thought Gauss's lemma was obvious by looking at it from two different directions, but both of those directions were incomplete. The first was the first half of the given proof, but I forgot to consider that the two factors in Z of n*f might not reduce to make factors of f in Z. The other thought I had was considering that polynomials in Q add no extra roots from polynomials in Z, considering all complex roots, but that doesn't directly address the problem at hand. It might make a good proof somehow, but I don't care enough to work out the missing bits.

2.3 & 3.1, due on September 11

Difficult:
    The concepts were mostly review, but it took me a while to see how the proof was done for the division algorithm. On a slightly deeper level, I don't understand why studying the "primes" of the polynomials over subfields and subrings of the complex numbers would be so useful to understanding their arithmetic.

Reflective:
    The trick to the proof to the remainder theorem is really cool. I wouldn't have thought of using a binomial expansion of y and alpha to produce q. It's the quirky things like that that really make proofs interesting to me. Proving mathematical truths is a hard business (NP-hard, I think), and thus these intuitive leaps are profound indeed.

2.1-2.2, due on September 9

Difficult:
    I found the "proof" to the Fundamental Theorem of Algebra to be very dense, and when I referred to the errata, I found that one of the only parts I understood was one of the errors. I did at least understand the winding numbers bit, but only because I remember it from complex analysis.

Reflective:
    I find it interesting how many uses polynomials have across all fields of mathematics. One thing that comes to mind is the concept of using polynomials to identify knots in knot theory. I have no idea what such polynomials mean or how to identify or generate them, but its strange connections like that that made me choose a math major.

1.0-1.4 due on September 6

Difficult:
    The most difficult part for me to understand was the historical development that lead to the suspicion that the quintic was unsolvable. I didn't see quite what was being permuted in the various formulas or how it implied radicals solved all the quadratics, cubics, and quartics.

Reflective:
    In the initial analysis, it was odd to me that mathematicians would use these non-constructive relations to show that quadratics, cubics, and quartics have solutions by radicals, when that fact was obvious by the preexisting formulas for finding roots. Then I thought about the nature of mathematical work. The mathematical landscape is not a expansive plain where the easiest way to see new land is to move forward, but is more akin to a hedge maze, where you often need to double back to go forward. In addition, mathematicians are not content merely to prove a statement and be done with it, but wish to see how it connects with other mathematical truths. From this viewpoint, reproving old statements with new methods is natural and desirable.
    Another thing that I found interesting is the fact that only completing the square is usually stated directly as a formula (the quadratic equation) while the methods for finding solutions to cubics and quartics are generally stated as a series of steps to be taken. I do not know whether this is for pedagogical reasons, historical reasons, or just because there isn't a good notation for getting the point across in a single formula. Even if the single equation form is unruly, I think I would prefer it to the set of steps given in the textbook and other sources I've read.
    I also found it vaguely amusing that Wednesday's lecture looked like it came straight from sections 1.1 and 1.2.

Introduction, due on September 6

Year in School:
    Senior

Major:
    Math

Math Taken Beyond Calculus:
    Math 313, Math 334, Math 341, Math 342, Math 352, Math 371

Why I am Taking This Class:
    I really enjoyed Math 371 and want to see a little further into where it goes.

Worst Math Teacher and Why:
    I haven't had a math teacher that has significantly affected my learning since the 3rd grade. My 3rd grade teacher was especially poor because she didn't appreciate my tendency to work out math problems in my own way.

What is Unique About Me:
    I love that there is so much variety in the world, even as I prefer to stay in my personal corner of it. As a result, I hate the hate, even the playful hate, involved in school pride.
    On a completely unrelated note: I have a bizarre affection for fixed-width fonts.

Good Alternate Office Hours:
    Anything on Tuesday or Thursday works for me.