A question of Kaplansky (Q1858284)

In 1941, Kurosh posed the question of whether an algebraic associative algebra over a field \(F\) was necessarily locally finite. A negative answer was given in 1962 by \textit{E. S. Golod} (on the basis of the universal construction due to Golod-Shafarevich), who constructed an infinite dimensional affine nil ring [see Izv. Akad. Nauk SSSR, Ser. Mat. 28, 273-276 (1964; Zbl 0215.39202)], and later, by Amitsur (unpublished), who constructed a primitive algebraic ring that is not locally finite. Amitsur's example gave an affirmative answer to the first half of a question posed by Kaplansky. The paper under review gives an affirmative answer to the second half of this problem by showing (constructively) the existence, over any \(F\), of an infinite dimensional primitive algebraic affine \(F\)-algebra.