Interesting properties of the partitions of a finite field $\mathbb F_q$ induced by the combination of involutions and trace maps are studied. The special features of involutions of the form $\frac{u}{z}$, $u$ being a fixed element of $\mathbb F_q$, are exploited to generate pseudorandom numbers, the randomness resting on the uniform distribution of the images of zero-trace elements among the sets of non-zero trace elements of $\mathbb F_q$.