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

@article{Bhargava2010OnAN, title={On a notion of "Galois closure" for extensions of rings}, author={Manjul Bhargava and Matthew Satriano}, journal={arXiv: Commutative Algebra}, year={2010} }

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.

#### 14 Citations

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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 show… Expand

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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 their… Expand

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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-module… Expand

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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 $A/R$. Many… Expand

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In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about:
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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 by… Expand

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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 an… Expand

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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. Constructions… Expand

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