On a notion of "Galois closure" for extensions of rings

  title={On a notion of "Galois closure" for extensions of rings},
  author={Manjul Bhargava and Matthew Satriano},
  journal={arXiv: Commutative Algebra},
We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions. 
Galois closures of non-commutative rings and an application to Hermitian representations
Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We showExpand
On the Galois closure of commutative algebras
In this thesis we prove several properties of the Galois closure of commutative algebras defined by Manjul Bhargava and Matthew Satriano. We also define some related constructions, and study theirExpand
To generalize the notion of Galois closure for separable field extensions, we devise a notion of G-closure for algebras of commutative rings R → A, where A is locally free of rank n as an R-moduleExpand
The Galois closure for rings and some related constructions
Let $R$ be a ring and let $A$ be a finite projective $R$-algebra of rank $n$. Manjul Bhargava and Matthew Satriano have recently constructed an $R$-algebra $G(A/R)$, the Galois closure of $A/R$. ManyExpand
FI-modules and stability for representations of symmetric groups
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on anExpand
On the Splitting Ring of a Polynomial
Let f(Z) = Zn − a1Zn−1 + … + (−1)n−1an−1Z + (−1)nan be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal If of R[X1,…, Xn] generated byExpand
Two inquiries about finite groups and well-behaved quotients
This thesis addresses questions in representation and invariant theory of finite groups. The first concerns singularities of quotient spaces under actions of finite groups. We introduce a class ofExpand
The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field
Abstract Let C be a smooth projective curve defined over the finite field F q ( q is odd) and let K = F q ( C ) be its function field. Removing one closed point C af = C − { ∞ } results in anExpand
A new discriminant algebra construction
A discriminant algebra operation sends a commutative ring $R$ and an $R$-algebra $A$ of rank $n$ to an $R$-algebra $\Delta_{A/R}$ of rank $2$ with the same discriminant bilinear form. ConstructionsExpand
Quartic Rings Associated to Binary Quartic Forms
We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava’s cubic resolvent rings.Expand


Arithmetic moduli of elliptic curves
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"Expand
Galois theory for schemes
Introduction 1–5 Coverings of topological spaces. The fundamental group. Finité etale coverings of a scheme. An example. Contents of the sections. Prerequisites and conventions. 1. Statement of theExpand
Higher composition laws III: The parametrization of quartic rings
In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields couldExpand
The moduli space of commutative algebras of finite rank
The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme $\frakB_n$ of finite type over $\Z$, with geometrically connected fibers. It is smooth if and onlyExpand
Representation Theory of Finite Groups: Algebra and Arithmetic
Dedication page Introduction Semisimple rings and modules Semisimple group representations Induced representations and applications Introduction to modular representations General rings and modulesExpand
Notes on Grothendieck topologies, fibered categories and descent theory
This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.
The Representation Theory of the Symmetric Group
1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetricExpand
Higher composition laws IV: The parametrization of quintic rings
rst three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involvingExpand
Hilbert schemes of 8 points
The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_nExpand
On the characteristic polynomial of a sum of matrices
A formula is proved which relates the coefficients of the characteristic polynomial of a sum of matrices ΣtiAi with the coefficients of the characteristics polynomials in the monomials At1 …At, t⩽n.Expand