Elements of large order in prime finite fields (Q2847608)

The results in this paper are related to a very difficult problem, namely, given a finite field \(\mathbb F_q\), how to quickly produce a generator of its multiplicative group \(\mathbb F^*_q\). The results in this paper are related to a work of \textit{J. von zur Gathen} and \textit{I. Shparlinski} [Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications, University of Augsburg, Germany, 1999. Berlin: Springer, 162--177 (2001; Zbl 1019.11036)], the results of which roughly spoken are that if \(F(x,y)\) is absolutely irreducible and \(F(x,0)\) is not a monomial, given a solution \((a,b)\) of \(F(x,y)=0\) (in the extension field) such that \(d=[\mathbb F_q(a), \mathbb F_q]\) is sufficiently large, then either \(a\) or \(b\) has a large multiplicative order (which are also quantified in the paper).