Elements of large order in prime finite fields

Authors

  • M.-C. Chang University of California, Riverside

Keywords:

multiplicative order, multiplicative group, finite fields, additive combinatorics

Abstract

Given $f(x,y)\in\mathbb Z[x,y]$ %such that the zero set of $f$ has with no common components with %those of $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, except $C(f)$ exceptions, the solutions $(x,y)\in\overline{\mathbb F}_p\times \overline{\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gtrsim \big(\frac{\log p}{\log\log p}\big)^{1/2},$ where ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline{\mathbb F}_p^*$. Moreover, for most $p p^{1/4+\epsilon(p)},$ where $\epsilon(p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty$. 10.1017/S0004972712000810

Author Biography

M.-C. Chang, University of California, Riverside

Professor of Mathematics, University of California at Riverside

Published

2013-07-23

Issue

Vol. 88 No. 1 (2013)

Section

Articles
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