In their seminal paper on translation planes (The Construction of Translation Planes from Projective Spaces, Journal of Algebra 1:85-102, 1964, https://doi.org/10.1016/0021-8693(64)90010-9), Bruck and Bose proved that every translation plane coordinatized by a quasifield that is finite-dimensional over its kernel can be represented in a projective space over the kernel. In this paper, the authors assert that this condition is satisfied by all known translation planes.
In the intervening years, have there been any published examples of quasifields that are infinite-dimensional over their kernel?
