I don't think I'll have that much more to say about Galois theory. But I think it's been a pretty good run. I don't think there's anywhere on the Internet where you'll find a better explanation of why you can't solve the quintic than what we've posted here over the last couple of months. At least, you won't find a more understandable explanation.
It's not that I'm smarter than the other people in the game. It's that I'm playing a different game than they are. In academia, the big thing in math is to be rigorous and abstract. What they value is exactness and economy, and if you ask them why something works that way, they say: which line of the proof do you not understand? I don't think that's what math is about.
The difference is most clear when you consider my emphasis on the question: what are the functions on five letters which map to each other under any permutation of those five letters? (That's the question Balarka answered last month on stackexchange which led to our correspondence in the last several blogposts.) Mathematicians don't ask this question. In doing Galois theory, they ask about the existence of normal subgroups. I would venture to say that the great majority of them don't even realize that the significance of those normal subgroups is precisely that they are associated with my "functions that map to each other under permutation." And by the way...it's not a one-to-one association. I didn't realize it at the time, but the function Balarka gave me, "Dummit's Function", is associated with the trivial subgroup consisting of exactly one element...the identity subgroup. So for S5, the resulting quotient group is just S5 itself. What it means is you won't find Dummit's Functions if you're looking for a normal subgroup to generate them.
What makes me think mathematicians don't look at things this way? I think I'm on pretty solid ground here. I would say that the fact that Dummit's Functions weren't even discovered until 1991 (!) is pretty good evidence that this way of thinking is outside the mainstream.
The point of all this is that the unsolvability of the quintic equation makes (almost!) intuitive sense if you play my game and look for functions of five letters that map to each other under permutation...but it becomes totally opaque if you abstract away all the concrete manifestations of the group action, and restrict yourself to talking about towers of normal subgroups. Even the well-known definition of a normal subgroup makes no intuitive sense...it's a subgroup for which "every left coset is also a right coset". What are we supposed to make of that? Mathematicians do that kind of thing all the time...they define two numbers p and q as being relatively prime if "there exists a and b such that ab-pq=1". That's a definition? It's no so harmful in this instance because we all know intuitively what it means for two numbers to be relatively prime, but in group theory, when you cascade one opaque definition like this on top of another, the whole subject rapidly becomes incomprehensible.
So at the end of the day, I'm pretty satisfied that I've explained what's really going on with the fifth degree equation. There may be a few fine points I haven't nailed down, but I'm not too worried about that. If you want to know why the quintic is unsolvable, I don't think you'll find a better place to start looking for answers than this blog right here.
Having gotten that out of the way, I think I'm going to return to my previous order of business, where I was posting my last-year's articles from the Jewish Post. I have to admit that my physics audience is not necessarily taking to this material with enthusiasm, but it's kind of important for me to get them out there. So let's see what's next...