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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 degree 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.
This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.
Let be an integral domain with the quotient field , an indeterminate over and an element of . The Bhargava ring over at is defined to be . In fact, is a subring of the ring of integer-valued polynomials over . In this paper, we aim to investigate the behavior of under localization. In particular, we prove that behaves well under localization at prime ideals of , when is a locally finite intersection of localizations. We also attempt a classification of integral domains ...
For and open in , let be the ring of real valued functions on with the first derivatives continuous. It is shown that for there is with and with . The function and its derivatives are not assumed to be bounded on . The function is constructed using splines based on the Mollifier function. Some consequences about the ring are deduced from this, in particular that .
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