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Scalar extension of quadratic lattices

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Department of MathematicsNagoya University
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Let E/F be a finite extension of algebraic number fields, OE, OF the maximal orders of E, F respectively. A classical theorem of Springer [6] asserts that an anisotropic quadratic space over F remains anisotropic over E if the degree [E: F] is odd. From this follows that regular quadratic spaces U, V over F are isometric if they are isometric over E and [E : F] is odd. Earnest and Hsia treated similar problems for the spinor genera [2, 3]. We are concerned with the quadratic lattices. Let L, M be quadratic lattices over OF in regular quadratic spaces U, V over F respectively.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 66 , May 1977 , pp. 139 - 149
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

References

[1] Borel, A., Introduction aux groupes arithmétiques, Hermann, Paris, 1967.Google Scholar
[2] Earnest, A. G. and Hsia, J. S., Springer-type theorems for spinor genera of quadratic forms, Bull. Amer. Math. Soc., 81 (1975), 942–943.CrossRefGoogle Scholar
[3] Earnest, A. G. and Hsia, J. S., Spinor genera under field extension (to appear in Acta Arith.).CrossRefGoogle Scholar
[4] Narkiewicz, W., Elementary and analytic theory of algebraic numbers, Warszawa, 1974.Google Scholar
[5] O’Meara, O. T., Introduction to quadratic forms, Springer-Verlag, Berlin, 1963.CrossRefGoogle Scholar
[6] Springer, T. A., Sur les formes quadratiques d’indice zéro, C. R. Acad. Sci., 234 (1952), 1517–1519.Google Scholar