bug in polredabs() (fwd)

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

In response to
> Message-Id: <19981204193944.K14565@io.txc.com>
> Date: Fri, 4 Dec 1998 19:39:44 -0500
> From: Igor Schein <igor@txc.com>
[...]
> So x^16+48 and x^16+3 generate the same number field,

No, they don't define the same field.  They define two distinct
fields which happen to have the same discriminant 2^48*3^15.

(If they did, then the field would also contain a 16th root of
(-48)/(-3) = 16, or in other words, a 4th root of 2.  But it doesn't
even contain a square root of 2 -- try to nffactor x^2-2 over the
result of nfinit(polredabs(y^16+3)).  The fields are quite hard to
tell apart otherwise -- a quick glance at the first 160 coefficients
of their Dedekind zeta functions suggests that they may be arith-
metically equivalent... any Galois representation experts at hand?
I'm afraid I've had too long a week at work to check this.)

That x^16+48 is difficult to work with is not surprising -- the
discriminant of the polynomial is divisible by a large power of
3 and by a huge power of 2, and the integer-base algorithm has
to figure out how much of each is due to the field discriminant
and how much to the index...  x^16-12*x^8+48 is marginally better
(2^108*3^15 instead of 2^124*3^15),  but still a lot harder than
x^16+3 with polynomial discriminant 2^64*3^15.

Cheers, Gerhard