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.
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