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 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 , then the root discriminant of is or so. However, by consulting Odlzyko, one sees that any totally real field with this root discriminant has degree at most . Hence the class number of 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 , one can compute by relating 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 (or so), which seems to prevent one from applying this to real cyclotomic fields for . (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.
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:
I’m not sure if this particular cut-out is at MoMA, though.
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 from to . Recently, I finally managed to complete the game:
Notice that the 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!
I proved Stickelberger’s theorem in class the other day — well, with one caveat. I proved that all the ideals 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?