From: "Malcolm Harper" Subject: Re: An eg of a PID which is not an ED Date: Wed, 27 Dec 2000 16:02:52 GMT Newsgroups: sci.math Summary: [missing] "Jan-Christoph Puchta" wrote in message news:3A49E6BB.21E491B3@arcade.mathematik.uni-freiburg.de... > Malcolm Harper wrote: > > > "Kyrre Kjoelner Andersen" wrote in message > > news:3A34A68B.B1C0CA99@rdg.ac.uk... [A question about Motzkin's proof that the ring of integers of Q(sqrt{-19}) is principal but not Euclidean] > > I can help with the "not ED" part of the argument. A good read on the > > subject is Samuel's _About Euclidean rings_, Journal of Algebra 19 (1971), 282--301. > > The PID-part can be done using Dirichlet's class number formula > and computing L(1, chi) numerically. > > > > > > The idea of the proof lies in the minimal algorithm for a Euclidean R, > > Samuel discusses this idea in detail. The proof goes through word for word > > for R when -19 is replaced by -43, -67 or -163. It is believed that these > > 4 fields are the only number fields with a ring of integers that is > > principal but not Euclidean (Weinberger proved this assuming a generalized > > Riemann hypothesis). The proof I give above fails outside the imaginary > > quadratic fields where R will have an infinite unit group. > > > > Oops, I think to remember a diploma thesis some years ago where it was proven > that the same is true for some cyclotomic field (I think 32th)! > Was this flawed, or is GRH wrong? > > JCP > Hopefully GRH is not wrong . Perhaps you are thinking of the result that while Q(zeta_{32}) has class number one, the absolute value of the norm does _not_ act as a Euclidean algorithm. I believe this result is due to H. W. Lenstra Jr. It appears in the first of his three Intelligencer articles on Euclidean number fields 1979--80. It may very well also be in his thesis which I believe was in the early 1970's. If it is some other result you are thinking of, I would be very interested in a reference. In my recent thesis I include a proof that Q(zeta_{32}) is Euclidean for an algorithm other that the norm. Thanks, Malcolm