Division algebras over - and -fields

Author:
Louis H. Rowen

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1607-1610

MSC (1991):
Primary 11R52, 12E15, 16K20, 16K50

Published electronically:
December 27, 2001

MathSciNet review:
1887005

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Abstract | References | Similar Articles | Additional Information

Abstract: Using elementary methods we prove a theorem of Rost, Serre, and Tignol that any division algebra of degree 4 over a -field containing is cyclic. Our methods also show any division algebra of degree 8 over a -field containing is cyclic.

**1.**Albert A.A.,*Noncyclic algebras*, Bull. Amer. Math. Soc.**38**(1932), 449-456.**2.**A. Adrian Albert,*Structure of algebras*, Revised printing. American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. MR**0123587****3.**S. A. Amitsur and D. Saltman,*Generic Abelian crossed products and 𝑝-algebras*, J. Algebra**51**(1978), no. 1, 76–87. MR**0491789****4.**M. Artin,*Brauer-Severi varieties*, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 194–210. MR**657430****5.**Marvin J. Greenberg,*Lectures on forms in many variables*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0241358****6.**Serge Lang,*On quasi algebraic closure*, Ann. of Math. (2)**55**(1952), 373–390. MR**0046388****7.**Rost M., Serre J.P., and Tignol J.P.,*The trace form of a central simple algebra of degree four*, paper in preparation.**8.**Jean-Pierre Tignol,*Sur les classes de similitude de corps à involution de degré 8*, C. R. Acad. Sci. Paris Sér. A-B**286**(1978), no. 20, A875–A876 (French, with English summary). MR**0498496****9.**Tsen C.,*Zur fStufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Korper*, J. Chinese Math. Soc.**1**(1936), 81-92.

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Additional Information

**Louis H. Rowen**

Affiliation:
Department of Mathematics & Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
rowen@macs.biu.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06277-3

Keywords:
Division algebra,
cyclic,
$C_{n}$-field

Received by editor(s):
November 16, 2000

Received by editor(s) in revised form:
January 3, 2001

Published electronically:
December 27, 2001

Additional Notes:
The author was supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities - Center of Excellence Program no. 8007/99-3

These results were discovered following conversations with David Saltman, to whom the author expresses his thanks. The author also thanks the referee for helpful comments.

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2001
American Mathematical Society