From: victor@watson.ibm.com (Victor S. Miller)
Newsgroups: sci.math.numberthy
Subject: "Nice" generators for Number Fields
Date: 12 May 92 21:00:23 GMT
Sender: NMBRTHRY@VM1.NoDak.EDU
Does anyone know of anything published on finding "nice" generators
for Number Fields. Specifically, if K=Q(\alpha), \alpha integral, is
"Nice" if none of the elements of the dual basis to {\alpha^k} is too
large. Alternatively, one would like none of the f'(\alpha_j) to be
too small, where f is the irreducible polynomial satisfied by \alpha,
and \alpha_j is the j-th conjugate.
Mahler ("An Inequality for the Discriminant of a Polynomial") gives a
general bound:
|f'(\alpha_j)| >= (m-1)^{-(m-1)/2} |D(f)|^{1/2} L(f)^{-(m-2)}
However, this is rather bad for certain polynomials: f(x) = x^n - a.
I would imagine that one could try to find a short vector in the
lattice given by the integers in the usual embedding into R^n.
Victor S. Miller
IBM, TJ Watson Research Center
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From: cohen@labri.greco-prog.fr (Henri COHEN [vauqueli])
Newsgroups: sci.math.numberthy
Subject: Re: "Nice" generators for Number Fields
Date: 13 May 92 13:35:38 GMT
There is indeed a well known "folklore" method on the subject
which is exactly as Victor suggests: finding short vectors in the
lattice given by the integers in the usual embedding into R^n.
I submitted a short paper joint with Diaz y Diaz to Math. Comp.
on the subject, and it was rejected on the grounds that it did
not contain any new mathematics, which is entirely true. However,
it is so useful that we decided to submit it elsewhere, and
it was finally published in the last issue of the new
"Seminaire de theorie des nombres de Bordeaux". For those who
want to try it, get the Pari/GP system from math.ucla.edu,
the function that you want is called POLRED. It works very nicely.
H. Cohen
cohen@alioth.greco-prog.fr
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