Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-19T02:27:03.086Z Has data issue: false hasContentIssue false

Sums of three integral squares in cyclotomic fields

Published online by Cambridge University Press:  17 April 2009

Chun-Gang Ji
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, China and Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, China, e-mail: cgji@amss.ac.cn
Rights & Permissions [Opens in a new window]

Abstract

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 m be an odd positive integer greater than 2 and f the smallest positive integer such that 2f ≡ 1 (mod m). It is proved that every algebraic integer in the cyclotomic field ℚ(ζm) can be expressed as a sum of three integral squares if and only if f is even.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Estes, D.R. and Hsia, J.S., ‘Exceptional integers of some ternary quadratic forms’, Adv. Math. 45 (1982), 310318.CrossRefGoogle Scholar
[2]Estes, D.R. and Hsia, J.S., ‘Sums of three integer squares in complex quadratic fields’, Proc. Amer. Math. Soc. 89 (1983), 211214.CrossRefGoogle Scholar
[3]Maass, H., ‘Über die Darstellung total positiver Zahlen des Köpers als Summe von drei Quadraten’, Abh. Math. Sem. Hansischen Univ. 14 (1941), 185191.CrossRefGoogle Scholar
[4]Moser, C., ‘Représentation de −1 par une somme de carrés dans certains corps locaux et globaux, et dans certains anneaux d'entiers algébriques’, C. R. Acad. Sci. Paris Ser. A-B 271 (1970), A1200A1203.Google Scholar
[5]Siegel, C.L., ‘Sums of mth powers of algebraic integers’, Ann. of Math. 46 (1945), 313339.CrossRefGoogle Scholar