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Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a primepower, $t$ is a positive integer. Let $f(x)$ be a polynomial over$\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, weconstruct a sufficient condition on $(p,t)$ which guarantees the existence of aprimitive normal pair $(\epsilon,f(\epsilon))$ such that$Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=a$,$Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=b$ and$N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=c$,$N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=d$ where $c,d\in\mathbb{F}_{p}$are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, wedemonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only$4$ possible exceptions where such pairs may not exist.

A primitive normal pair in a finite field with prescribed traces and norms