Relations with conjugate numbers over a finite field (Q1881797)

Let \(K\) be a field, \(\overline K\) its algebraic closure and \(n\geq 2\). The author considers the following two equations: \(\beta = k_1 \alpha_1 + k_2 \alpha_2 +\cdots k_n \alpha_n\) with given \(k_j \in K\setminus \{0\}\) and \(\beta = \alpha_1^{k_1} \alpha_2^{k_2} \cdots \alpha_n^{k_n} \) with given non-zero integers \(k_j\), where (in both cases) \(\alpha_1,\ldots, \alpha_n\in \overline K\) denote conjugate elements over \(K\). The main results says that for finite \(K\) and for all \(\beta \in \overline K\) both equations have more than one solution. The proof makes use of properties of cyclic extentions and of Hilbert's Theorem 90.