Splitting fields and separability

Author:
Mark Ramras

Journal:
Proc. Amer. Math. Soc. **38** (1973), 489-492

MSC:
Primary 16A16

DOI:
https://doi.org/10.1090/S0002-9939-1973-0314888-X

MathSciNet review:
0314888

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Abstract: It is a classical result that if is a complete discrete valuation ring with quotient field , and if is perfect, then any finite dimensional central simple -algebra can be split by a field which is an unramified extension of . Here we prove that if is *any* regular local ring, and if contains an -order whose global dimension is finite and such that is central simple over , then the existence of an ``-unramified'' splitting field for implies that is -separable. Using this theorem we construct an example which shows that if 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 which has no -unramified splitting field.

**[1]**M. Auslander and D. A. Buchsbaum,*On ramification theory in noetherian rings*, Amer. J. Math.**81**(1959), 749-765. MR**21**#5659. MR**0106929 (21:5659)****[2]**M. Auslander and O. Goldman,*The Brauer group of a commutative ring*, Trans. Amer. Math. Soc.**97**(1960), 367-409. MR**22**#12130. MR**0121392 (22:12130)****[3]**M. Ramras,*Orders with finite global dimension*, Pacific J. Math. (to appear). MR**0349758 (50:2251)****[4]**O. F. G. Schilling,*The theory of valuations*, Math. Surveys, no. 4, Amer. Math. Soc., Providence, R.I., 1950. MR**13**, 315. MR**0043776 (13:315b)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0314888-X

Keywords:
Central simple algebra,
splitting field,
separable,
-unramified,
maximal order,
global dimension

Article copyright:
© Copyright 1973
American Mathematical Society