Related papers: Interpretations and differential Galois extensions

We prove that any Galois extension of commutative rings with normal basis and abelian Galois group of odd order has a self dual normal basis. Also we show that if S/R is an unramified extension of number rings with Galois group of odd order…

Let $k$ be a differential field having an algebraically closed field of constants, $E$ be a strongly normal extension of $k$, and $k^0$ be the algebraic closure of $k$ in $E.$ We prove for any intermediate differential field $k\subset…

We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local…

This paper concerns the description of holomorphic extensions of algebraic number fields. We define a hyperbolized adele class group for every number field K Galois over Q and consider the Hardy space H[K] of graded-holomorphic functions on…

Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is…

This paper deals with the Weak Inverse Galois Problem which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to explain how…

Given an arbitrary field $F$, we describe all Galois extensions $L/F$ whose Galois groups are isomorphic to the group of upper triangular unipotent 4-by-4 matrices with entries in the field of two elements.

We study stable rationality of conic bundles $X$ over $\mathbb{P}^1$ defined over non-closed field $k$ via the cohomology of the Galois group of finite field extension $k'/k$ with action on the geometric Picard lattice of $X$.

The aim of this paper is to give a new result of the differential Galois theory of linear ordinary differential equations. In particular, we compute differential Galois group for special type non-resonant Fuchsian system.

Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the…

Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also…

We present a simple proof of the fundamental theorem of Galois theory, which establishes a correspondence between the intermediate fields of a finite Galois extension and the subgroups of its Galois group. The proof is based on the…

We consider a dynamical system consisting of a pair of commuting power series, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$-adic integers. Assuming that each point of the dynamical system…

In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}(T)$ of group $G$ with $E\cap\overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the…

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable…

Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if…

One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois…

In \cite{GQ2008} R. Gow and R. Quinlan have cast a new look on the endomorphism algebra of a $K$-vector space $V$ of dimension $n$ assuming that $K$ has a Galois extension $L$ of degree $n$. In this approach the $K$-space $L$ may serve as a…

We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the…