In Brief

The start of the academic year has a habit of bringing forth distractions, not least of all to someone as disorganized as me. So here are a few remarks in brief.

The class number of \mathbf{Q}(\zeta_{151})^{+} is one.

John Miller, a student of Iwaniec at Rutgers, wrote the following nice paper, which improves upon a previous result of Schoof. One technique that is useful in computing the class numbers of fields with small discriminant is to make use of the Odlyzko bounds. Here’s a typical example. If K =\mathbf{Q}(\zeta_{37})^{+}, then the root discriminant of K is 30 or so. However, by consulting Odlzyko, one sees that any totally real field with this root discriminant has degree at most 40. Hence the class number of K is either one or two, and it is easy to rule out the second possibility by using genus theory. More generally, whenever one has an a priori bound on h^{+}, one can compute h^{+} by relating h^{+} to the index of the circular units (Schoof did this in a previous paper.) This trick only works if the root discriminant of the totally real field is at most 60 (or so), which seems to prevent one from applying this to real cyclotomic fields for p > 67. (There’s always a bound on the class group by Minkowski, but that is a terrible bound.) The idea behind this paper is that Odlyzko’s bound can be improved if one in addition knows that certain primes of small norm are principal. And since one has explicit fields, it is possible to show that the relevant ideals are principal by exhibiting explicit elements with the appropriate norm. I can’t quite tell how lucky the author was to find such elements (he searches for cyclotomic elements expressible as a small number of roots of unity), but it works! Perhaps, a postiori, it is useful that these fields do actually turn out to have class number one.

Matisse cut-outs

The NYT reports on an exhibit of Matisse cut-outs at MoMA. I have a particular soft spot for these works. I was particularly struck by a cut-out I once saw at an exhibit at the Centre Pompidou, so much so that I painted a replica (almost full sized) on the wall of my rental apartment in Cambridge:

My version

Original

I’m not sure if this particular cut-out is at MoMA, though.

256

Thanks to DS, I was hooked on 2048 for far too long. I eventually got bored trying to get the 16384 tile, and moved on to the more compact 256 instead. The latter game is slightly less random in that only 2s are created on each turn. The highest possible score is 7172, which is obtained when one ends up with (in any configuration) the powers of 2 from 2 to 512. Recently, I finally managed to complete the game:

Notice that the 512,256,128 tiles are not along the same edge (I think it must be theoretically possible to finish in that way, but it would be harder). Unfortunately, having reached this point, it has not cured me of my addiction. Curse you, Savitt!

Stickelberger’s Theorem.

I proved Stickelberger’s theorem in class the other day — well, with one caveat. I proved that all the ideals \mathfrak{q} of prime norm are annihilated by the Stickelberger ideal. This certainly implies the result, because the class group is generated by such ideals. This follows, for example, by the Cebotarev density theorem applied to the Hilbert class field (which was my argument in class). But then I worried that this was an anachronistic argument, and indeed Stickelberger’s theorem was a solidly 19th century result. So what did Stickelberger do?

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9 Responses to In Brief

  1. Speaking of fields with class number one, have you seen this preprint by Darren Long and Morwen Thistlethwaite:

    http://www.math.ucsb.edu/~long/pubpdf/LenstraHurwitz.pdf

    ?

    • I saw from Davide’s email list that Darren gave a talk a U Chicago, so I looked up the paper on my iPhone on the way to work. It was nice, but I think it unlikely that *in practice* it could ever produce infinitely many class number one fields.

  2. John Miller says:

    Long and Thistlethwaite’s paper is quite interesting. They mention that to have even a chance of producing a prime clique big enough to show class number 1, the root discriminant must be less than 16*pi*e. This raises the question: Although we believe that fields with class number 1 are quite plentiful, do we have any reason to expect there are infinitely many fields of class number 1 with bounded root discriminant? Towers of fields with class number 1 should have increasing root discriminant; and on the other hand most towers with bounded root discriminant are built out of Hilbert class fields of fields with nontrivial class groups.

    • The only clearly convincing heuristics (to me) about class numbers concern families of fields of fixed degree (so the discriminant necessarily tends to infinity). The problem of understanding fields with bounded root discriminant (where the bound > Odlyzko) on the other hand seems very quite difficult. It’s *possible* that one could have a tower of fields all with class number one and the same root discriminant (if the corresponding Galois groups are perfect), although it doesn’t seem particularly plausible on some sketchy heuristic grounds.

  3. You probably shouldn’t try DIV.

    http://alexfink.github.io/dive/

    I’m not sure if it’s more or less evil to hand that one to a number theorist than to a mere mortal. (In my experience it was just too hard to be dangerous.)

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