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On the class number of a unit lattice over a ring of real quadratic integers

Published online by Cambridge University Press:  22 January 2016

Yoshio Mimura
Yoshio Mimura*
Affiliation:
Department of Mathematics, Kobe University, Nada-ku, Kobe 657, Japan
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Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 92 , December 1983 , pp. 89 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1983

References

[1] Dzewas, J., Quadratsummen in reell-quadratischen Zahlkörpern, Math. Nachr., 21 (1960), 233–284.Google Scholar
[2] Kneser, M., Klassenzahlen definiter quadratischen Formen, Arch. Math., 8 (1957), 241–250.CrossRefGoogle Scholar
[3] Nebelung, O., p-adische Darstellungsdichten binärquadratischer Formen, Dissertation, Ulm 1978.Google Scholar
[4] O’Meara, 0. T., Introduction to quadratic forms, Berlin-Göttingen-Heidelberg, Springer-Verlag, 1963.CrossRefGoogle Scholar
[5] Peters, M., Einklassige Geschlechter von Einheitsformen in totalreellen algebraischen Zahlkörpern, Math. Ann., 226 (1977), 117–120.CrossRefGoogle Scholar
[6] Pfeuffer, H., Einklassige Geschlechter totalpositiver quadratischer Formen in totalreellen algebraischen Zahlkörpern, J. Number Theory, 3 (1971), 371–411.Google Scholar
[7] Pfeuffer, H., Quadratsummen in totalreellen algebraischen Zahlkörpern, J. reine angew. Math., 249 (1971), 208–216.Google Scholar
[8] Pfeuffer, H., Über die reelle Spiegelungsgruppe und die Klassenzahl der sechsdimensionalen Einheitsform, Arch. Math., 31 (1978), 126–132.Google Scholar
[9] Pfeuffer, H., On a conjecture about class numbers of totally positive quadratic forms in totally real algebraic number fields, J. Number Theory, 11 (1979),Google Scholar
[10] Pohst, M., Mehrklassige Geschlechter von Einheitsformen in total reellen algebraischen Zahlkörpern. J. reine angew. Math., 262/263 (1973), 420–435.Google Scholar
[11] Salamon, R., Die Klassen im Geschlecht von über Z[ ], Arch. Math., 20 (1969), 523–530.Google Scholar