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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Counting Algebraic Numbers with Large Height I
verfasst von : David Masser, Jeffrey D. Vaaler
Erschienen in: Diophantine Approximation
Verlag: Springer Vienna
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Let ℚ denote the field of rational numbers,
an algebraic closure of ℚ, and
H
:
the absolute, multiplicative, Weil height. For each positive integer
d
and real number
$$ \mathcal{H} \geqslant 1 $$
, it is well known that the number
of points α in
having degree
d
over ℚ and satisfying
$$ H\left( \alpha \right) \leqslant \mathcal{H} $$
is finite. This is the one-dimensional case of Northcott’s Theorem [
8
] (see also [5, page 59]). The systematic study of the counting function
, and that of related functions in higher dimensions, was begun by Schmidt [
10
]. It is relatively easy to prove the existence of a positive constant
C = C(d)
such that
(1)
and also the existence of positive constants
c = c(d)
and
$$ \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) $$
such that
(2)