Related papers: Generators for abelian extensions of number fields

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in…

In this paper we obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a…

For a finite valued field extension $(L/K,v)$ we describe the problem of find sets of generators for the corresponding extension $\mathcal O_L/\mathcal O_K$ of valuation rings. The main tool to obtain such sets are complete sets of (key)…

For each positive integer $n$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of the finite field $\mathbb F_q$ with $q$ elements, where $q$ is a prime power. It is known that for arbitrary $q$ and $n$, there exists an element…

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

Let $A$ be an infinite set. Let $\Omega(A)$ be the algebra over $A$ where every constant is a fundamental constant and every finitary function is a fundamental operation. We shall give a method of representing any algebra $\mathcal{L}$ in…

The notion of normal elements for finite fields extension has been generalized as k-normal elements by Huczynska et al. [3]. The number of k-normal elements for a fixed finite field extension has been calculated and estimated [3], and…

This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

Algebraic methods are used to construct families of unramified abelian extensions of some families of number fields with specified Galois groups.

In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully…

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

In this work we present some arithmetic properties of families of abelian $p$--extensions of global function fields, among which are their generators and their type of ramification and decomposition.

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show…

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$…

We study the height of generators of Galois extensions of the rationals having the alternating group $\mathfrak{A}_n$ as Galois group. We prove that if such generators are obtained from certain, albeit classical, constructions, their height…

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…

We consider Abelian extensions of global symmetries of the form $A \to G \to K$, with $A$ finite (and similar higher-group structures). For a quantum field theory $\mathcal{T}$ with symmetry $G$, we compare gauging $G$ directly with gauging…

Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…