Related papers: On Ring Class Fields of Number Rings

We compute the spinor class field for a genus of orders, in a central simple algebra of higher dimension, that are intersections of two maximal orders. In particular, we compute the number of spinor genera in a genus of such orders, as the…

We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multiplications by the elements of $K^*$. We also…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the…

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$,…

A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant…

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_q$ and finite cyclic rings $\mathbb{Z}/p^r\mathbb{Z}$, in the…

Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K^m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and…

The ring of ad\`eles of a global field and its group of units, the group of id\`eles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we…

Already Dedekind and Weber considered the problem of counting integral ideals of norm at most $x$ in a given number field $K$. Here we improve on the existing results in case $K/\mathbb Q$ is abelian and has degree at least four. For these…

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical…

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such…

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 $K$,…

In this paper, we define an ordering relation for a set of complex numbers, and research the properties and theorems of the ordering, solve some simple complex inequalities with the ordering.

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that…

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to…

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in \'etale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of…

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains…