Abstract
AbstractLet
$m>1$
and
$\mathfrak {d} \neq 0$
be integers such that
$v_{p}(\mathfrak {d}) \neq m$
for any prime p. We construct a matrix
$A(\mathfrak {d})$
of size
$(m-1) \times (m-1)$
depending on only of
$\mathfrak {d}$
with the following property: For any tame
$ \mathbb {Z}/m \mathbb {Z}$
-number field K of discriminant
$\mathfrak {d}$
, the matrix
$A(\mathfrak {d})$
represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
Publisher
Canadian Mathematical Society
Reference12 articles.
1. Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent;Harron;Algebra Number Theory,2021
2. [10] Mantilla-Soler, G. and Rivera, C. , Real ${S}_n$ number fields and trace forms. Preprint, 2019. arXiv:1812.03133v3
3. The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields
4. The shapes of pure cubic fields
5. The shape of
-number fields;Mantilla;Ramanujan J.,2016