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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Splitting fields and separability

Author: Mark Ramras
Journal: Proc. Amer. Math. Soc. 38 (1973), 489-492
MSC: Primary 16A16
MathSciNet review: 0314888
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Abstract: It is a classical result that if $ (R,\mathfrak{M})$ is a complete discrete valuation ring with quotient field $ K$, and if $ R/\mathfrak{M}$ is perfect, then any finite dimensional central simple $ K$-algebra $ \Sigma $ can be split by a field $ L$ which is an unramified extension of $ K$. Here we prove that if $ (R,\mathfrak{M})$ is any regular local ring, and if $ \Sigma $ contains an $ R$-order $ \Lambda $ whose global dimension is finite and such that $ \Lambda /\operatorname{Rad} \Lambda $ is central simple over $ R/\mathfrak{M}$, then the existence of an ``$ R$-unramified'' splitting field $ L$ for $ \Sigma $ implies that $ \Lambda $ is $ R$-separable. Using this theorem we construct an example which shows that if $ R$ is a regular local ring of dimension greater than one, and if its characteristic is not 2, then there is a central division algebra over $ K$ which has no $ R$-unramified splitting field.

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Keywords: Central simple algebra, splitting field, separable, $ R$-unramified, maximal order, global dimension
Article copyright: © Copyright 1973 American Mathematical Society

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