  The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$
and any finite field $\mathbb F_q$, there exists an element $\alpha\in \mathbb
F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In
this paper, we prove some variations of this result, completing the proof of a
conjecture proposed by Anderson and Mullen (2014). Our results also imply the
existence of elements of $\mathbb F_{q^n}$ with multiplicative order
$(q^n-1)/2$ and prescribed trace over $\mathbb F_q$.
