Gerhard Niklasch on Sat, 5 Dec 1998 02:26:04 +0100 (MET) |
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Re: bug in polredabs() (fwd) |
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