Let k ≥ 1 and $$f{_1}, \ldots, f{_r} \in {\mathbb F}_{q^k}(x)$$ be a system of rational functions forming a strongly linearly independent set over a finite field $${\mathbb F}_q$$ . Let $$\gamma_1, \ldots, \gamma_r \in {\mathbb F}_q$$ be arbitrarily prescribed elements. We prove that for all sufficiently large extensions $${\mathbb F}_{q^{km}}$$ , there is an element of prescribed order such that ${\rm Tr}_{{\mathbb F}_{q^{km} }/{\mathbb F}_q}(f_i(\xi))=\gamma_i$ for $i=1, \ldots, r$, where is the relative trace map from $${\mathbb F}_{q^{km}}$$ onto ${\mathbb F}_q$. We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.