Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T05:53:52.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Gyclotomic Division Algebras

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin
R. A. Mollin*
Affiliation:
Queen's University, Kingston, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.

Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1074 - 1084
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Connell, I., The Stufe of number fields, Math. Z. 124 (1972), 20–22.Google Scholar
2. Goldstein, L. J., Analytic number theory, (Prentice Hall, Englewood Cliffs, N.J., 1971).Google Scholar
3. Janusz, G., Algebraic number fields, (Academic Press, New York, 1973).Google Scholar
4. Mollin, R., Uniform distribution and the Schur subgroup, J. Algebra $ (1976), 261–277.Google Scholar
5. Mollin, R., Uniform distribution and real fields, J. Algebra 43 (1976), 155–167.Google Scholar
6. Mollin, R., Algebras with uniformly distributed invariants, J. Algebra 44 (1977), 271–282.Google Scholar
7. Mollin, R., U(K) for a quadratic field K, Communications in Algebra 4 (1976), 747–759.Google Scholar
8. Mollin, R., Generalized uniform distribution of Hasse invariants, Communications in Algebra 5 (1977), 245–266.Google Scholar
9. Mollin, R., Her stein1 s conjecture, automorphisms and the Schur group, Communications in Algebra 6 (1978), 237–248.Google Scholar
10. Mollin, R., The Schur group of a field of characteristic zero, Pacific J. Math. 76 (1978), 471–478.Google Scholar
11. Mollin, R., Uniform distribution classified, Math. Z. 165 (1979), 199–211.Google Scholar
12. Mollin, R., Induced p-elements in the Schur group, Pacific J. Math. 90 (1980), 169–176.Google Scholar
13. Mollin, R., Splitting fields and group characters, J. reine angew Math. 315 (1980) 107–114.Google Scholar
14. Mollin, R., Cyclotomic splitting fields (to appear: Canadian Mathematical Bulletin).Google Scholar
15. Reiner, I., Maximal orders, (New York, Wiley Interscience, 1972).Google Scholar
16. Yamada, T., The Schur subgroup of the Brauer group, Lecture Notes in Mathematics 397 (Springer, Berlin, Heidelberg, New York, 1974).Google Scholar