论文标题
在数字字段中分类和标记不可或缺的理想
Sorting and labelling integral ideals in a number field
论文作者
论文摘要
我们定义了标记和订购数字字段不可或缺的理想的方案，包括主要理想作为特殊情况。我们定义的顺序仅取决于选择每个字段$ k $的一元不可减少的积分定义多项式，我们首先为每个字段定义其在belabas之后的唯一降低的定义多项式。我们对$ K $的主要理想集合定义了总订单，然后将其扩展到$ k $的所有非零积分理想的集合。该订单使我们能够给出$ n.i $的唯一标签，其中$ n $是其标准，$ i $是所有理想$ n $的理想列表中理想的索引。我们的理想标签方案具有几个不错的特性：对于给定的规范，主要的理想总是首先出现，并且鉴于规范的分解，norm $ n $和标签的理想之间的两次射击是可以在多项式时间内计算的。 我们这样做的动机是有一种定义明确，简洁的方法，以对LMFDB等数据库进行分类和标记理想。我们已经实施了算法，这些算法在Sage，Magma和Pari中实现了该方案。
We define a scheme for labelling and ordering integral ideals of number fields, including prime ideals as a special case. The order we define depends only on the choice of a monic irreducible integral defining polynomial for each field $K$, and we start by defining for each field its unique reduced defining polynomial, after Belabas. We define a total order on the set of prime ideals of $K$ and then extend this to a total order on the set of all nonzero integral ideals of $K$. This order allows us to give a unique label of the form $N.i$, where $N$ is its norm and $i$ is the index of the ideal in the ordered list of all ideals of norm $N$. Our ideal labelling scheme has several nice properties: for a given norm, prime ideals always appear first, and given the factorisation of the norm, the bijection between ideals of norm $N$ and labels is computable in polynomial time. Our motivation for this is to have a well-defined and concise way to sort and label ideals for use in databases such as the LMFDB. We have implemented algorithms which realise this scheme, in Sage, Magma and Pari.
