On fundamental sets over a finite field (Q1064350)

Let \(\alpha \in GF(p^ n)\). The authors define \(A_{\alpha}=\{\alpha,\alpha +1,...,\alpha +p-1\}\), \({}^*A_{\alpha}=\cup^{p-1}_{l=1}A_{l\alpha}\), \(\bar A_{\alpha}=\cup^{n-1}_{l=0}{}^*A_{\alpha}^{p^ l}\), and \(\alpha\sim\beta\) iff \(\bar A_{\alpha}=\bar A_{\beta}\). The equivalence relation \(\sim\) partitions the field \(GF(p^ n)\) into fundamental classes. To study the fundamental classes the authors discuss the solutions of the equations of the form \(x^{p^ m}=ax+b\). In the last section they investigate the number of fundamental classes for some finite fields.