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Constructions of Brauer-Severi Varieties and Norm Hypersurfaces

Published online by Cambridge University Press:  20 November 2018

Ming-Chang Kang
Ming-Chang Kang*
Affiliation:
National Taiwan University, Taipei, Republic of China
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Let k be any field, A a central simple k-algebra of degree m (i.e., dimk A = m2). Several methods of constructing the generic splitting fields for A are proposed and Saltman proves that these methods result in almost the same generic splitting field [8, Theorems 4.2 and 4.4]. In fact, the generic splitting field constructed by Roquette [7] is the function field of the Brauer- Severi variety Vm(A) while the generic splitting field constructed by Heuser and Saltman [4 and 8] is the function field of the norm surface W(A). In this paper, to avoid possible confusion about the dimension, we shall call it the norm hypersurface instead of the norm surface.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 2 , 01 April 1990 , pp. 230 - 238
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Artin, M., Brauer-Severi varieties, in Brauer groups in ring theory and algebraic geometry, Springer LNM 917 (Springer-Verlag, Berlin, 1982).Google Scholar
2. Dieudonné, J., Sur une generalisation du group orthogonal a quatre variables, Archiv. der Math. (1948), 282–287.Google Scholar
3. Draxl, P.K., Skew fields (Cambridge University Press, Cambridge, 1983).Google Scholar
4. Heuser, A., Uber den Funktionenkorper der Normfiache einer zentral einfachen Algebra, J. Reine Angew. Math. 301 (1978), 105–113.Google Scholar
5. Jacobson, N., Structure groups and Lie algebras of Jordan algebras of symmetric elements of associative algebras with involution, Advance in Math. 20 (1976), 106–150.Google Scholar
6. Marcus, M. and Moyls, B.N., Linear transformations on algebras of matrices, Can. J. Math. 11 (1959), 61–66.Google Scholar
7. Roquette, P., On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting field of algebras, Math. Ann. 150 (1962), 411—439.Google Scholar
8. Saltman, D., Norm polynomials and algebras, J. Algebra 62 (1980), 333–345.Google Scholar
9. Serre, J.P., Local fields, English translation, Springer GTM 67 (Springer-Verlag, New York, 1979).Google Scholar
10. Waterhouse, W.C., Linear maps preserving reduced norms, Linear Algebra and Its Appl. 43 (1982), 197–200.Google Scholar