On the Kurosh problem for algebras of polynomial growth over a general field

Lenagan and Smoktunowicz (2007) [LS], (see also Lenagan, Smoktunowicz and Young (in press) [LSY]) gave an example of a nil algebra of finite Gelfand–Kirillov dimension. Their construction requires a countable base field, however. We show that for any field k and any monotonically increasing function f(n) which grows super-polynomially but subexponentially there exists an infinite-dimensional finitely generated nil k-algebra whose growth is asymptotically bounded by f(n). This construction gives the first examples of nil algebras of subexponential growth over uncountable fields.