Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T11:32:17.682Z Has data issue: false hasContentIssue false
  • English
  • Français

Sums of m-th powers in p-adic rings

Published online by Cambridge University Press:  26 February 2010

C. P. Ramanujam
C. P. Ramanujam
Affiliation:
Tata Institute of Fundamental Research, Bombay
Get access

Extract

Let A be a complete discrete valuation ring of characteristic zero with finite residue field, and for any integer m > 1, let Jm (A) be the subring of A generated by the m-th powers of elements of A. We will prove that any element of Jm (A) is a sum of at most 8m5m-th powers of elements of A. We will also prove a similar assertion when the residue field of A is only assumed to be perfect and of positive characteristic, with the number Γ(m) of summands depending only on m and not on A.

Type
Research Article
Information
Mathematika , Volume 10 , Issue 2 , December 1963 , pp. 137 - 146
Copyright
Copyright © University College London 1963

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Birch, B. J., “Waring's problem in algebraic number fields”, Proc. Cambridge Phil. Soc., 57 (1961).CrossRefGoogle Scholar
1.Körner, O., “Über Mittelwerte trigonometrischer Summen und ihre Anwendung in algebraischen Zahlkörpern”, Math. Annalen, 147 (1962).CrossRefGoogle Scholar
1.Siegel, C. L., “Generalisation of Waring's problem to algebraic number fields”, American J. of Math., 66 (1944).CrossRefGoogle Scholar
1.Stemmler, R. M., “The easier Waring problem in algebraic number fields”, United States Office of Naval Research Technical Report N.R., 043–194 (1959).Google Scholar
1.Tornheim, L., “Sums of n-th powers in fields of prime characteristic”, Duke Math. J., 4 (1938).CrossRefGoogle Scholar