I learned something new about the fifth-degree equation over the last few days. The other day I told you that the reason you can't solve the quintic has something to do with the fact that there exists no set of functions on five letters which "map to themselves under all permutations". Or something like that. Well, it turns out there is...only there isn't. Here's what I learned.
I had constructed these three functions on four letters (corresponding to the four roots of a quartic equation:)
AB + CD = p
AC + BD = q
AD + BC = r
We can see that no matter how you re-shuffle A, B, C and D, you get back the same three functions p, q and r. I said that the reason you can't solve the quintic was that there is no comparable set of four functions on five letters (A, B, C, D and E) which is similarly preserved on taking permutations of letters.
And that's true. But what I didn't know is that you can construct a set of six functions on those five letters which have the desired property...sort of.
I had almost concluded that there were no such functions, and I posted a question on stackexchange.com here to see if I was right. I wasn't. One Balarka Sen pointed out that there were a number of such functions, including one called Dummit's Resolvent, which seems to have been discovered only twenty years ago by a math professor from the U of Vermont. Dummit's functions look something like this:
AA(BE + CD) + BB(CA + DE) + CC(DB + EA) + DD(EC + AB) + EE(AD + BC)
There are actually six of these functions...you can generate the other five by suitable permutations of the A's, B's and C's, as Sen explains in his answer to my question on stackexchange.
Now, what people say next is a little misleading: they say that these six functions generate a sixth-degree equation, and therefore they're not going to help you solved a fifth-degree. That's not quite true. It might be a sixth-degree equation of a simpler form than the fifth-degree which you started from. It's certainly not a sixth-degree equation of the most general type, because the permutations of the roots cannot be more complicated than the permutations of the five letters which generated them. In other words, the Galois Group of the sixth degree equation cannot be bigger than S5, the permutation group on five elements. But could it be a simpler Galois Group? Maybe it could...
But here's where group theory actually gives us some guidance. I said that Dummet's functions are similar by analogy to my p, q and r which gave me a third-degree resolvent to the general fourth-degree equation. But in one crucial respect they fail to do what p, q and r do. It's true that unlike my p, q and r, Dummits functions are more numerous than the five roots that generated them. That's a problem, but it's not decisive. The real problem is that there are no (non-trivial) permutations on the five letters which leave all six of Dummit's functions in place.
I'm going to leave it for another day to tell you why I consider this circumstance to be critical. Suffice it to say (for now) that there are indeed permutations of A, B, C and D which leave my p, q, and r in place, and the set of all such permutations forms what they call a normal subgroup of S4. But before we get there, I have to tell you one more thing. In one way or another, I've been working on this problem off and on for most of forty years. It's a great problem and I don't think it gets its due in the undergraduate curriculum....I've talked about that previously. So when I posted the question the other day on stackexchange, asking about the existence of these functions, I wasn't sure what I'd get, and I was happy to get a pretty clear answer from that fellow Sen. Now here's the kicker...it turns out that out Balarka Sen is a thirteen-year-old kid from India.
Thirteen years old????
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3 comments:
First off, nice post. I like how you are explaining the insolvability of quintics. Usually textbooks jumps on to technicalities of group theiry instead of presenting a new idea, so definitely these blog posts are a bit off-the-chain.
Now, I'd like to add a few pointers :
It is indeed untrue what people says about resolvent sextics. They can help a lot while solving quintics. As for whether Galois group can be simpler than that of quintics, in solvable cases : yes! In fact, for solvable quintics, they are reducible over Q[x], and the real tough part is to show vice-versa, which gives you a criterion for solvable quintics. As for nonsolvable quintics, negative but interesting enough they are a bit simpler than that of general sextics. In fact, that is what helps one to solve such sextic by elliptic functions and order 1 theta functions, whereas in general, sextics are solved in terms of hyperelliptic and double thetas.
Now, for the normal subgroup you are referring to, I believe it is V4, i.e., Viergrouppe? Otherwise a generator of A4 permutes the roots.
(PS : I was invited here by Green from stackexhange)
(PPS : Too bad blogspot doesn't support latex. I believed wordpress didn't too until I saw Tao's. How much black magic did it took?!)
I'm delighted to hear from you, Balarka. You're right about Latex. I do my equations in Paint (!) and then paste them into my blogpost as images. Or I do them in MS Word and snip them as images. (But of course that doesn't work in the Comments field.)
As for the Vierguppe, I looked it up on Wikipedia, and yes, that's what I was thinking of. You'll see the Wikipedia article is quite explicit about how it lets you create a cubic which is the stepping stone to the quartic.
I don't know why I had such a hard time figuring out those resolvents. I'd actually tried multiplying out A + wB + w^2C... (I think that's called the Lagrange resolvent) and throwing away the symmetric pieces, which basically works for the third and fourth degree equations, but I guess I got lost. I take some consolation in that the Dummit functions seem to be brand new.
I have no idea how you can possibly know all this stuff. Are you interested in physics at all? Send me an email sometime, marty at "onforeignsoil" dot com.
It is nice to hear back from you too. As for the LaTeX, you should use [http://www.codecogs.com/latex/eqneditor.php]. It automatically renders and compiles LaTeX snippets and you can even downlaod them as images.
Yes, Lagrange usually gets pretty complicated for higher degrees, that's why I omitted it in the stackexchange question.
As for physics, no. I simply hate physics.
I'll surely send you email sometimes. My address is balarka2000(at)gmail dot com.
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