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.