Re: bug in polredabs() (fwd)

[Gerhard Niklasch <nikl@mathematik.tu-muenchen.de>]

In further response to
> > Message-Id: <19981204193944.K14565@io.txc.com>
> > Date: Fri, 4 Dec 1998 19:39:44 -0500
> > From: Igor Schein <igor@txc.com>
subsequent to
> Message-Id: <199812050059.BAA14280@pchelwig1.mathematik.tu-muenchen.de>
> Date: Sat, 5 Dec 1998 01:59:18 +0100 (MET)
from yours truly:

> [...]
> > So x^16+48 and x^16+3 generate the same number field,
> 
> No, they don't define the same field.

Cute.  x^8+3 and x^8+48 already define distinct fields (quadratic
extensions of the same quartic field -- totally complex of discriminant
432 --) which, however, both have class number 1, the same regulator
24.0787745..., and the same order of the torsion unit subgroup (6th
roots of unity).

> They define two distinct
> fields which happen to have the same discriminant 2^48*3^15.

...and the same class number (1), and presumably the same regulator,
although this is hard to tell at default realprecision.  (The bnf[8][3]
components differ by more than the computed regulators - almost 3%.)

By the way, does anybody have a few handy examples of pairs of
arithmetically equivalent fields of degree 7 or 8 in which the
smallest nontrivial ideal norm is larger than 3?  I'm quite
pleasantly surprised at this handy example in degree 8 where
at least there are no ideals of norm 2. :^)  I've got just one
pet item of curiosity I've long wanted to check out on such
specimens, and which is of no interest at all in the presence
of ideals of norm 2...

Enjoy, Gerhard