Related papers: Arithmetic of Division Fields

In this paper we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the…

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised reductive group schemes, such as $L$-groups and $C$-groups. We show that the corresponding deformation rings are complete…

The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem…

We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple…

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions $L/K$ is a natural next step. One must…

Given an abelian variety over a field with a discrete valuation, Grothendieck defined a certain open normal subgroup of the absolute inertia group. This subgroup encodes information on the extensions over which the abelian variety acquires…

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient…

One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the…

We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number…

We classify cuts in (totally) ordered abelian groups $\g$ and compute the coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with…

We study normal extensions with Galois group Hol($C_8$) that are unramified over a complex quadratic subfield. The Galois group is either the semi-dihedral group or the modular group of order $16$. We present an explicit construction of…

Let F be a Henselian valued field with char(F) = p and D a semi-ramified, "not strongly degenerate" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras,…

For finite Galois extension fields defined by odd degree irreducible polynomials over algebraic integer ring, we observe "Reciprocity Law" through Jacobian Variety by embedding all roots of the polynomials into 2-torsion points of Jacobian…

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of…

For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the…

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups.…

Let $A$ be a 2-dimensional abelian variety defined over a number field $K$. Fix a prime number $\ell$ and suppose $\#A(\mathbb{F}_p) \equiv 0 \pmod{\ell^2}$ for a set of primes $\mathfrak{p} \subset \mathcal{O}_K$ of density 1. When…

We consider continuous representations of the Galois group G of a number field K taking values in the completion C of an algebraic closure A of the field of l-adic numbers. We give a construction of irreducible representations of G in…

It is shown that a Hopf algebra over a field admitting a Galois extension separable over its subalgebra of coinvariants is of finite dimension. This answers in the affirmative a question posed by Beattie et al. in [{\it Proc. Amer. Math.…

Current techniques for formally verifying circuits implemented in Galois field (GF) arithmetic are limited to those with a known irreducible polynomial P(x). This paper presents a computer algebra based technique that extracts the…