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An elementary approach is shown which derives the value of the Gauss sum of acubic character over a finite field $\mathbb F_{2^s}$ without usingDavenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if$s$ is even its value is $-(-2)^{s/2}$).

Gauss Sums of the Cubic Character over $GF(2^m)$: an elementary derivation