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An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field F2s without using Davenport-Hasse’s theorem (namely, if s is odd the Gauss sum is 1, and ifs is even its value is ( 2) s=2 ).

Gauss Sums of the Cubic Character over GF(2m): an Elementary Derivation

Gauss Sums of the Cubic Character over GF(2^m): an Elementary Derivation