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On the decomposition of a field as a tensor product

Published online by Cambridge University Press:  18 May 2009

P. M. Cohn
P. M. Cohn
Affiliation:
Bedford College, Regent's Park, London NW1 4NS
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The following two results in the theory of division algebras are well known and easily proved, for an arbitrary commutative field k (cf. for example [3, Chapter 10]).

(i) The tensor product of two central division algebras over k of coprime degrees is again a division algebra.

(ii) Every central division algebra over k is a tensor product of division algebras of prime power degrees.

It is natural to ask whether corresponding results hold for commutative fields. The answers are not hard to find but (as far as I am aware) have not appeared in print before; since they throw some light on the nature of tensor products they seemed worth recording.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 20 , Issue 2 , July 1979 , pp. 141 - 145
Copyright
Copyright © Glasgow Mathematical Journal Trust 1979

References

REFERENCES

1.Bourbaki, N., Algèbre Ch.V (Hermann, Paris 1950).Google Scholar
2.Brown, K. A., Zero-divisors in group rings, Bull. London Math. Soc. 8 (1976), 251–256.CrossRefGoogle Scholar
3.Cohn, P. M., Algebra II (J. Wiley, Chichester1 1977).Google Scholar
4.Farkas, D. R., Miscellany on Bieberbach group algebras, Pacific J. Math. 59 (1975), 427–435.CrossRefGoogle Scholar
5.Farkas, D. R. and Snider, R. L., Ko and Noetherian group rings, J. Algebra 42 (1976), 192–198.CrossRefGoogle Scholar
6.Hall, M. JrThe theory of groups (Macmillan, New York 1959).Google Scholar
7.Huppert, B., Endliche Gruppen I (Springer, Berlin 1967).CrossRefGoogle Scholar