# Existence of rational primitive normal pairs over finite fields

Document Type : Research Paper

Authors

1 Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi, India

2 Scientific Analysis Group, Defence Research and Development Organisation, Metcalfe House, Delhi, India

3 Mathematics, Assistant Professor, S.S. Govt. P.G. College, Tigaon, Faridabad, Haryana, India

Abstract

For a finite field $𝔽_{q^n}$ and a rational function $f=\frac{f_1}{f_2} \in 𝔽_{q^n}(x)$, we present a sufficient condition for the existence of a primitive normal element $\alpha \in 𝔽_{q^n}$ in such a way $f(\alpha)$ is also primitive in $𝔽_{q^n}$, where $f(x)$ is a rational function in $𝔽_{q^n}(x)$ of degree sum $m$ (degree sum of $f(x)=\frac{f_1(x)}{f_2(x)}$ is defined to be the sum of the degrees of $f_1(x)$ and $f_2(x)$). Additionally, for rational functions of degree sum 4, we proved that there are only $37$ and $16$ exceptional values of $(q,n)$ when $q=2^k$ and $q=3^k$ respectively.

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