Existence of rational primitive normal pairs over finite fields

For a finite field 𝔽_{q^n} and a rational function f=\frac{f_۱}{f_۲} \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_۱(x)}{f_۲(x)} is defined to be the sum of the degrees of f_۱(x) and f_۲(x)). Additionally, for rational functions of degree sum ۴, we proved that there are only ۳۷ and ۱۶ exceptional values of (q,n) when q=۲^k and q=۳^k respectively.

کلیدواژه ها:

finite field ، Primitive Element ، Normal Element ، character