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