论文标题
数字字段中的规范关系和计算问题
Norm relations and computational problems in number fields
论文作者
论文摘要
对于有限的组$ g $，我们在组代数$ \ mathbb q [g] $中引入了规范关系的概括。我们为存在这种关系提供了必要和充分的标准，并将其应用于与Galois组$ g $的代数数字字段正常扩展的子场的算术不变性之间的关系。在算法方面，这导致了基于子场的算法，用于计算整数，$ s $单位组和班级组。对于$ s $单位组计算，这将使子场中的相应问题的多项式时间缩短。我们计算GRH下的大数字段的班级组，以及循环组磁场的类数量的新的无条件值。
For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group $G$. On the algorithmic side this leads to subfield based algorithms for computing rings of integers, $S$-unit groups and class groups. For the $S$-unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.
