The author proves that for any primitive element \(b\) of \(\mathbb F_q\) there exist primitive polynomials of degree \(3\) and \(4\) with zero trace and norm \(b\) with the only exceptions \(n=3\) and \(q=4\) or \(7\). Combining this with earlier results of \textit{S. Fan} and \textit{X. Wang} [Finite Fields Appl. 15, No. 6, 682--730 (2009; Zbl 1218.11109)] for \(n\geq 5\), \textit{S. D. Cohen} and \textit{S. Huczynska} [Acta Arith. 109, No. 4, 359--385 (2003; Zbl 1074.11064)]for \(n=4\) and \(a\neq 0\) and [Trans. Am. Math. Soc. 355, No. 8, 3099--3116 (2003; Zbl 1022.11064)] for \(n=3\) and \(a\neq 0\) this completes the proof of the following result: Let \(b\) be a primitive element of \(\mathbb F_q\) and \(a\in\mathbb F_q\). Then for any integer \(n\geq 3\), there exists a primitive polynomial of degree \(n\) over \(\mathbb F_q\) with trace \(a\) and norm \(b\) with the only exceptions when \(n=3\), \(a=0\), and \(q=4\) or \(7\).