Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T17:34:56.510Z Has data issue: false hasContentIssue false
  • English
  • Français

The Schur Subgroup of a p-Adic Field

Published online by Cambridge University Press:  20 November 2018

Eugene Spiegel  and
Allan Trojan
Eugene Spiegel
Affiliation:
University of Connecticut, Storrs, Connecticut
Allan Trojan
Affiliation:
Atkinson College, York University, Downsview, 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. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only if

where c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 300 - 303
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Albert, A. A., Structure of algebras, A.M.S. Colloquium Publications, vol. XXIV, Providence, R.I., (1964).Google Scholar
2. Bernard, and Schacher, , The Schur subgroup II, J. of Algebr. 22 (1972), 378–385.Google Scholar
3. Berman, S. D., Representations of finite groups over an arbitrary field and over rings of integers, Trans. Amer. Math. Soc. 64 (1967), 148–215.Google Scholar
4. Curtis, C. and Reiner, I., Representation theory of finite groups and associative algebras, (Interscience, New York, 1962).Google Scholar
5. Deuring, M., Algebrem, (2nd. ed., Springer-Verlag, Berlin, 1968).Google Scholar
6. Perlis, S. and Walker, G., Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420–426.Google Scholar
7. Serre, J.-P., Corps locaux, (Hermann, Paris, 1968).Google Scholar
8. Van der Waerden, B. L., Algebra, (vol. 2, 5th ed., Ungar, New York, 1967).Google Scholar
9. Yamada, T., Characterization of the simple components of the group algebras over the p-adic number field, J. Math. Soc. Japa. 23 (1971), 295–310.Google Scholar
10. Yamada, T., The Schur subgroup of a 2-adic field, J. Math. Soc. Japa. 26 (1974), 168–179.Google Scholar
11. Yamada, T., The Schur subgroup of the Brauer group, Lecture Notes in Mathematics 397, (Springer-Verlag, Berlin, Heidelberg, New York, 1974).Google Scholar