Primitive Element Pairs with a Prescribed Trace in the Quartic Extension of a Finite Field

In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\alpha$ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.Comment: 14 page

Primitive values of rational functions at primitive elements of a finite field

Given a prime power $q$ and an integer $n\geq2$, we establish a sufficient condition for the existence of a primitive pair $(\alpha,f(\alpha))$ where $\alpha \in \mathbb{F}_q$ and $f(x) \in \mathbb{F}_q(x)$ is a rational function of degree $n$. (Here $f=f_1/f_2$, where $f_1, f_2$ are coprime polynomials of degree $n_1,n_2$, respectively, and $n_1+n_2=n$.) For any $n$, such a pair is guaranteed to exist for sufficiently large $q$. Indeed, when $n=2$, such a pair definitely does {\em not} exist only for 28 values of $q$ and possibly (but unlikely) only for at most $3911$ other values of $q$.Comment: 12 page

On existence of some special pair of primitive elements over finite fields

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha \in \mathbb{F}_{p^k}$ such that $f(\alpha)$ is also primitive in $\mathbb{F}_{p^k}$, where $f(x) \in \mathbb{F}_{p^k}(x)$ is a quotient of polynomials with some restrictions. We explicitly determine the values of $k$ for which such a pair exists for $p=2,3,5$ and $7$

Existence of Primitive Pairs with Prescribed Traces over Finite Fields

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs $(\alpha, f(\alpha))$ in $F$ such that for any prescribed $a, b$ in $E=\mathbb{F}_q$, Tr$F/E (\alpha) = a$ and Tr$F/E (\alpha^{-1}) = b$. Further, for every positive integer $n$, such a pair definitely exists for large enough $(q,m)$. The case $n = 2$ is dealt separately and proved that such a pair exists for all $(q,m)$ apart from at most $64$ choices