The shape of cyclic number fields
The shape of cyclic number fields
复制标题
循环数域的形状
DOI：
10.4153/s0008439522000546
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Guillermo Mantilla
中科院分区：
文献类型：
--
作者：
Wilmar Bolanos;Guillermo Mantilla
抽象Let
$m>1$
和
$\mathfrak {d} \neq 0$
是整数，使得
$v_{p}（\mathfrak {d}）\neq m$
对于任意素数p，我们构造一个矩阵
$A（\mathfrak {d}）$
身型限重
$（m-1）\times（m-1）$
只取决于
$\mathfrak {d}$
具有以下属性：对于任何驯服
$ \martbb {Z}/m \martbb {Z}$
- 判别式的数域K
$\mathfrak {d}$
矩阵
$A（\mathfrak {d}）$
表示K的整数迹零形式的格拉姆矩阵。特别地，我们得到了驯服循环数域的积分迹零形式由域的度和判别式决定。此外，如果除了上述假设之外，我们考虑真实的数域，则形状也由度和判别式确定。
Abstract Let
$m>1$
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.