Related papers: Counting ideals in abelian number fields

Let L be an abelian number field of degree n with Galois group G. In this paper we study how to compute efficiently a normal integral basis for L, if there is at least one, assuming that the group G and an integral basis for L are known.

Let $I_1\subset I_2\subset\dots$ be an increasing sequence of ideals of the ring $\Bbb Z[X]$, $X=(x_1,\dots,x_n)$ and let $I$ be their union. We propose an algorithm to compute the Gr\"obner base of $I$ under the assumption that the…

In this paper, we describe the higher even $K$-groups of the ring of integers of a number field in terms of class groups of an appropriate extension of the number field in question. This is a natural extension of the previous collective…

The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right…

Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or…

In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of…

The number of ad-nilpotent ideals of the Borel subalgebra of the classical Lie algebra of type B_n is determined using combinatorial arguments involving a generalization of Dyck-paths. We also solve a similar problem for the untwisted…

We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree $n$ polynomials $f$…

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction…

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of…

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

Given a graded ideal $I$ in a polynomial ring over a field $K$ it is well known, that the number of distinct generic initial ideals of $I$ is finite. While it is known that for a given $d\in\N$ there is a global upper bound for the number…

We prove a result relating the Jacobian ideal and the generalized test ideal associated to a principal ideal in $R=k[x_1,...,x_n]$ with $[k:k^p]<\infty$ or in $R=k[[x_1,...,x_n]]$ with an arbitrary field $k$ of characteristic $p>0$. As a…

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all…

For a number field $K$ let $\mathcal{S}_K$ be the maximal subgroup of the multiplicative group $K^\times$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group $\mathcal{S}_K$ can be seen as an…

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero…

In our earlier article~\cite{CanSakran} we initiated a study of the complement-finite submonoids of the group of integer points of a unipotent linear algebraic group. In the present article, we continue to develop tools and techniques for…