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On the Compacity of the Orthogonal Groups

Published online by Cambridge University Press:  22 January 2016

Takashi Ono
Takashi Ono*
Affiliation:
Mathematical Institute, Nagoya University
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It is a well known fact on Lorenz groups that a quadratic form f is definite if and only if the corresponding orthogonal group On(R∞, f) where R∞ is the real number field, is compact. In this note, we shall show that the analogue of this holds for the case of the p-adic orthogonal group On(Rp, f), where Rp is the rational p-adic number field, as a special result of the more general statement on the completely valued fields.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 7 , June 1954 , pp. 111 - 114
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1954

References

[ 1 ] Artin, E., Algebraic numbers and algebraic functions, Princeton (1951).Google Scholar
[ 2 ] Dieudonné, J., Sur les groupes classiques, Act. Sci. et Ind., 1040 (1948).Google Scholar
[ 3 ] Witt, E., Theorie der Quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937).Google Scholar
[ 4 ] Dieudonné, J., Sur les groupes orthogonaux rationnels à trois et quatre variables. Compt. rend., t. 233 (1951).Google Scholar