Power Sums over Finite Subspaces of a Field

  • Authors:

  • Nigel P. Byott;Robin J. Chapman

  • Affiliations:

  • School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, United Kingdomf1E-mail: N.P.Byott@ex.ac.uk, R.J.Chapman@ex.ac.ukf1;School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE, United Kingdomf1E-mail: N.P.Byott@ex.ac.uk, R.J.Chapman@ex.ac.ukf1

  • Venue:
  • Finite Fields and Their Applications
  • Year:
  • 1999

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Abstract

LetVbe a finite additive subgroup of a fieldKof characteristicp0. We consider sums of the formS"h(V:@a)=@?"v"@?"V(v+@a)^hforh=0 and @a@?K. In particular, we give necessary and sufficient conditions for the vanishing ofS"h(V; @a), in terms of the digit sum in the base-pexpansion ofh, in the case thatVhas indexpinK. The proof involves the polynomialf"V(x)=@?"v"@?"V(x-v). We defineV^@?=f"V(K). In the case thatKis finite, the polynomialf"V"^"@?can be viewed as a generalised trace Tr"V:K-Vwhich coincides with the usual trace ifVis a subfield ofK.