On the number of integral ideals of a number field

The angular distribution of integral ideal numbers with a fixed norm in quadratic extensions

Erdős and Hall studied the angular distribution of Gaussian integers with a fixed norm. We generalize their result to the angular distribution of integral ideal numbers with a fixed norm in any quadratic extension.

On the logarithmic probability that a random integral ideal is $\mathscr A$-free

This extends a theorem of Davenport and Erdös on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and provide a formula for the logarithmic density of the set of so-called $\mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $\mathscr A$.

Sorting and labelling integral ideals in a number field

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…

Ideal Class Group Algorithms in the Ring of Integral Quaternions

An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the algorithms for ideal manipulation. However, this is not the only option, as an ideal can also be represented by its generating set. Moreover, in certain quadratic orders of the ring of integral quaternions, every ideal can be represented by a single quaternion, called a pseudo generator. In this paper, we develop algorithms which allow us to manipulate ideals using s…

Bass 序的轨道积分与理想类幺半群

Orbital integrals and ideal class monoids for a Bass order

本文针对 Bass 序证明了分式理想个数的闭式导子公式，并给出过度序的显式参数化，结合局部分类与轨道积分得到加权质量公式及几何定理。

正特征下互质w-的S-整数和S-整理想的计数

Counting w-coprime S-integers and S-integral ideals in positive characteristic

本文结合解析方法、Riemann-Roch定理和Weil定理，在正特征函数域中计数互质w-的S-整数和S-整理想。