A functional version of Hilbert's theorem 90 (Q1177269)

Let \(F\) be a finite extension field of \(\mathrm{GF}(q)\) and \(f(x_1,\ldots,x_n)\), a rational function with coefficients from \(F\). Suppose that \(f\) is such that for every finite extension \(E\) of \(F\), if \((a_1,\ldots,a_n)\in E^n\) and \(f(a_1,\ldots,a_n)\) is defined, then \(\operatorname{Tr}_{E/\mathrm{GF}(q)}f(a_1,\ldots,a_n)=0\). Then, it is proved in this paper, there exists a rational function \(g(x_1,\ldots,x_n)\) with coefficients in \(F\) such that \[ f(x_1,\ldots,x_n) = g(x_1,\ldots,x_n) - g(x_1,\ldots,x_n)^q. \] Actually, two different proofs are given. The first is a self-contained elementary proof, while the second is based on a theorem of \textit{S. Lang} and \textit{A. Weil} [Am. J. Math. 76, 819--827 (1954; Zbl 0058.27202)].