They were named by Piotr Blass in 1977 after Oscar Zariski who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational.
(In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
For p = 2 and for p = 3 the answer to the above problem is negative as shown in 1977 by Piotr Blass in his University of Michigan Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978.
Kentaro Mitsui (2014) announced further examples giving a negative answer to Zariski's question in every characteristic p>0 .
His method however is non constructive at the moment and we do not have explicit equations for p>3.