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...
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4 comments:
Just a note : As I have indicated in MSE, there are other function that Dummit's ones, and one of them found by Malfatti in 1771.
As a remark, I agree with you that these kind of explanation isn't really what mathematicians prefer, but again these kind of explanation is what mathematicians give and should give, no?
PS : I invite you to join a forum I am heavily active in : it's mymathforum (MMF, in short). It's not exactly for discussion of physics, but you may find one or two mathematical question that you may be interested in to leave an answer or two. The link is http://www.mymathforum.com/, if you join there, let me know your username via mail.
Yes, you're right about those functions. I didn't mean to make it look like Dummit's function was unique, but that's probably how it reads. I still think it's amazing that such a simple function should have been obscure for so long.
Your note about Malfatti's function is consistent with what I'm saying in a way, because it's 250 years old. My approach to to this problem is a lot more in tune with the 18th century outlook than anything you'll find today.
In retrospect the existence of these functions should have been obvious to me just from looking at the Lagrange Resolvent which I looked at many times without knowing its name, and even multiplied out to get its fifth power. I tried throwing away the symmetric pieces to see what was left, and simply didn't recognize that I had generated a resolvent function because I got confused by the counting. There are 120 orderings of the five letters, but the starting point of the cycle doesn't matter, so there are only 24 to count. What I didn't realize was that the four permutations of the fifth roots of unity ALSO collapsed when you took the fifth power...so that reduces your count to six functions, which obviously have to map to themselves under permutation.
In other words, it's obvious (where A,B,C etc are the roots and w is a fifth root of unity) that the fifth power of
A + wB + w^2C....
gives the same function as
B + wC + w^2D....
but I didn't realize it also gave the same function as
A + w^2B + w^4C....
...or do they? I think I was wrong. I thought I had a mechanical way of collapsing the Lagrange resolvents down to six functions, but after all I don't think I have.
Hmm....
Okay, NOW I've got it...I think! I think I can mechanically generate Dummit's resolvents by starting with a typical Lagrange resolvent, of which there are one hundred and twenty...but when you take the fifth power, you reduce them to twenty-four because the cyclic ones are multiples of the fifth root of unity. Now...here's the trick! you take those twenty-four and group them into six groups of four based on the permutations of the fifth roots of unity...like I said above,
A + wB + w^2C...
is related to
A + w^2B + w^4C...
Now...when you add together those groups of four, you get a bunch of symmetric stuff which you throw away, and you get a bunch of stuff in powers of w which cancels each other out...but you're left with a residue of terms, which I believe are exactly Dummit's resolvent. I'll have to check that...
If this really works, then it's all the more amazing that no one did it until 1991.
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