  We study the class of Bernstein algebras that are algebraic, in the sense
that each element generates a finite-dimensional subalgebra. Every Bernstein
algebra has a maximal algebraic ideal, and the quotient algebra is a
zero-multiplication algebra. Several equivalent conditions for a Bernstein
algebra to be algebraic are given. In particular, known characterizations of
train Bernstein algebras in terms of nilpotency are generalized to the case of
locally train algebras. Along the way, we show that if a Banach Bernstein
algebra is algebraic (respectively, locally train), then it is of bounded
degree (respectively, train).
  Then we investigate the Kurosh problem for Bernstein algebras: whether a
finitely generated Bernstein algebra which is algebraic of bounded degree is
finite-dimensional. This problem turns out to have a closed link with a
question about associative algebras. In particular, when the barideal is nil,
the Kurosh problem asks whether a finitely generated Bernstein-train algebra is
finite-dimensional. We prove that the answer is positive for some specific
cases and for low degrees, and construct counter-examples in the general case.
  By results of Yagzhev, the Jacobian conjecture is equivalent to a certain
statement about Engel and nilpotence identities of multioperator algebras. We
show that the generalized Jacobian conjecture for quadratic mappings holds for
Bernstein algebras.
