24.1-24.3, due on December 11

Difficult:
    I can't follow any of the given proofs.

Reflective:
    I think its interesting that all of these proofs come down to the fact that a sequence of integers that goes to zero in the limit must always be zero after some point.

22.3-22.4, due on December 9

Difficult:
    The instructions for calculating the Galois group of an arbitrary polynomial are absurd. You might as well crack RSA by factoring.

Reflective:
    I was most interested in the concept of the discriminant. Rather than doing a difficult process to find out exactly what the Galois group is, just rely a simple one to find out if it is in the alternating group or not.

22.1-22.2, due on December 6

Difficult:
    Learning the content wasn't so difficult, but I have no idea what motivated using phi and psi to calculate a Galois group.

Reflective:
    I find it interesting that a result that is supposed to be important and far reaching, namely the fundamental theorem of Galois theory, relies on such a difficult operation, namely finding Galois groups of polynomials, to be applied directly.

21.5-21.7, due on December 4

Difficult:
    The proofs in sections 21.5-21.6 all fit together nicely, but as was mentioned in class, they don't really imply theorem 21.3

Reflective:
    I find the table of cyclotomic polynomials in section 21.7 quite elegant. The fact that they all seem to have coefficients in the set {-1, 0, 1} is stated, but I find it interesting that they all seem to follow a simple pattern as well.

21.3-21.4, due on December 2

Difficult:
    I got lost at the end of the proof of theorem 21.2, most of page 243, specifically.

Reflective:
    It's not exactly clear to me how this result is so close to proving Galois theory; I do remember needing roots of unity, but how it leads to finding the rest is beyond me. Nevertheless, it is interesting that the history of mathematics could have taken a different turn if Vandermonde was more rigorous in his assertion that roots of unity had non-trivial radical expressions.

21.1-21.2, due on November 26

Difficult:
    Ironically, I understood everything that was going on until they starting including Galois groups, at which point I became entirely lost.

Reflective:
    I did not expect it to be so easy (relatively speaking) to find a meaningful radical expression for non-real n-th roots of unity.

20.1.-20.2, due on November 25

Difficult:
    The corollaries in section 20.2 were straight forward, but I couldn't follow the group exponent stuff before it. I also had trouble following theorem 20.2. What did they mean let p=p^n? how can you just say that?

Reflective:
    I find it interesting that the classification of finite fields is so simple. Perhaps there is more interesting structure in the non-commutative case, but it's surprising how few conditions you need to severely limit the structure of what remains.

18.4-18.5, due on November 22

Difficult:
    I was lost for all of 18.4

Reflective:
    It was interesting that every formula for finding roots is based on symmetries of the said roots.

Exam 2 Prep, due on November 18

Important:
    I think the most important bit is the Fundamental Theorem of Galois Theory, and how it is different in the complex numbers than in general fields, although we didn't quite finish the general case.

Expected:
   I expect a lot of finding Galois groups and proving solvability and finding normal subgroups.

Needed:
    I need a better grip on the content of FToGT, as well as the important lemmas that surround it.

17.5-17.6, due on November 15

Difficult:
    The Frobenius map made sense. Everything afterwards that relied on it did not.

Reflective:
    I am glad that the Frobenius map is the only way to mess up separability for fields. I bet it's a lot worse with non-commutative rings.

17.1-17.4, due on November 13

Difficult:
    I read the sections, but I couldn't get anything from 17.2 or 17.3 to stick.

Reflective:
    I at first found it strange that learning math from a book would be so much harder than from a teacher, until I realized that there are at least three qualities that teachers have that makes them superior to books. Teachers can respond to a lack of knowledge if you ask, they can perceive missing knowledge when you do not, and they force you to slow down enough to comprehend all of the material.

16.3-16.5, due on November 11

Difficult:
    The notation used for theorem 16.9 threw me a bit, making it hard to follow. Also, the proof for lemma 16.11 seems less than adequate.

Reflective:
    It almost feels more natural to be in general rings and fields than to be in C or Q, now that I'm a little further into it. I suppose the rigor of 371 made it so for me.

16.1-16.2, due on November 8

Difficult:
    This was all basically review from 371, so nothing too hard here. I was momentary confused by axiom M1, until I saw the note that greater generality was not necessary in this context. I didn't remember ideals from 371, but they were adequately defined here.

Reflective:
    I am worried about generalizing Galois theory to more general fields, as I have not fully internalized the theorems from chapter 12, and the added complexity of a need for separability will not help my case.

15.2-15.3, due on November 6

Difficult:
    There wasn't really anything difficult in these sections. All the hard work was in the lead up to this. I suppose if I must put something, I don't understand the point of writing all this down, as I learn so readily in the classroom, and so slowly on my own.

Reflective:
    I would be more impressed by Klein's unification achievement if I hadn't already heard of it in a previous book I read on group theory.

15.1, due on November 4

Difficult:
    I didn't quite follow the logic of the proof for lemma 15.4, where they say that the auto-morphism tau implies that beta-ij is radical. Also, I got lost in the proof of lemma 15.7, and the proof of theorem 15.3 wasn't terribly easy either

Reflective:
    I found the definition of a radical extension very pleasant for some reason. I guess I just enjoy seeing a intuitive concept reduced to a concise symbolic expression.

14.3, due on November 1

Difficult:
    The logic of the first lemma was entirely lost on me, and I got lost in the proof for Cauchy's theorem.

Reflective:
    I was looking at the homework due tomorrow, and I thought for a second that 14.5 was in the homework. I was quite excited because it was manifestly obvious that those groups would be solvable since they were built using extensions of order 2, so it would just be Z2 all the way down. It was disappointing to see that the homework skipped from 14.4 to 14.6 for that reason.

14.2, due on October 30

Difficult:
    I had trouble following the bit of the major proof where they showed that any normal subgroup of A5 and above contains a 3-cycle.

Reflective:
    Amazing that the proof for the impossibility of a general solution by radicals comes from showing A5 and above are simple.

14.1, due on October 28

Difficult:
    I could not follow any of the proofs.

Reflective:
    I suppose solubility of groups has to do with the existence of solutions to related polynomials, yes?

13.1, due on October 23

Difficult:
    The example seemed straight-forward, but I was disconcerted by the immediate recognition of the Galois Group as D8, as I have little experience in such identifications.


Reflective:
    With this example, along with yesterday's lecture, everything looks to be falling into place in my mind. The full beauty of the theory is beginning to become apparent, and I remember why I am a math major.

12.1, due on October 21

Difficult:
    There are so many ideas chasing each other around in these few pages that I can hardly hope to keep track of them all. I think I might need to reread the entire book that leads up to this.

Reflective:
    Well, we have the Fundamental Theorem of Galois Theory now. How are we going to use it to crack the tough nuts of algebra?

11.2, due on October 18

Difficult:
    In stark contrast to section 11.1, I could not follow much of anything in this section. This is especially bad considering the importance of the result derived from this work.


Reflective:
    I suppose I should be impressed by corollary 11.11 or theorem 11.12, but I was too confused by what led up to it to really appreciate it.

11.1, due on October 16

Difficult:
    This section was incredibly short, as well as straight-forward. The only difficulty is my continued neglect of ideas behind the commutativity diagrams. I really need to get a handle on those.

Reflective:
    Reducing the size of what to search for by making a monomorphism rather than an automorphism, then get back an automorphism anyway is quite neat.

10.1, due on October 14

Difficult:
    I couldn't quite follow the second half of theorem 10.5 where m > n is shown to be impossible.

Reflective:
    Example 10.7.2 was interesting because it showed the difference between Q(exp(2*pi*i/n)) and Q(m*exp(2*pi*i/n)), where m^n is in Q but m is not. Where as the first has degree n-1, the second has degree n.

9.2-9.3, due on October 11

Difficult:
    I have no idea what the "g" refers to in the proof of Theorem 9.9.


Reflective:
    I find it interesting that one of the test questions would have been trivially easy with a bit of reading ahead.

9.1, due on October 9

Difficult:
    I didn't quite follow the argument for lemma 9.5.


Reflective:
    I'm glad that there is a specific topic revolving around the idea that since a field extension doesn't always split its associated minimal polynomial e.g.: Q(2^(1/3)) and t^3-2, there is name for the field extension that does.

Exam 1 Prep, due on October 7

Important:
    I think the most important topic we've studied is field extensions, since that is what the everything revolved around from chapter 4 on.

Expected:
    I expect the exam to be predominately proofs about field extensions. The rest is probably polynomial stuff.

Needed:
    I understand things fine. What I need is to make sure I have everything in my head. To help with this, I'm going to collect all the definitions, lemmas, and theorems in the text into one sheet and memorize that. Reviewing how the major theorems we're proven might also be helpful. I just hope I don't need Cardano's method memorized.

8.1-8.6, due on October 4

Difficult:
    The reverse inclusion ideas of 8.6 confuse me. I almost have as handle on it, but I have little confidence in my current hunches. Also, I'm having trouble following the arguments for why gamma + delta and gamma * delta are in Q in section 8.3.


Reflective:
    The most interesting thing I found in this was the tangential remark that all geometries can be thought of as a theory of relevant invariants mentioned at the beginning of section 8.5.

7.2, due on October 2

Difficult:
    This section follows very simply from section 7.1, but as I have yet to wrap my mind around that as a result of sleeping through Monday's class, my understanding of all of the impossibility proofs are castles without foundations, and cannot stand on their own.


Reflective:
    All of the impossibility proofs rest on the idea that minimal polynomials of the constructions have to have a degree that is a power of 2 to be possible. This is a powerful idea, as exemplified by the relative brevity of all the proofs in section 7.2. It is neat that such an odd idea as using algebra to represent geometric constructions lends itself to such brief proofs.

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.