  In this paper we prove that all irrational numbers from totally real cubic
number fields are well approximable by rationals (i.e. the partial quotients in
the continued fraction expansion of such a number are unbounded). This settles
the long standing open question of whether or not well approximable algebraic
numbers exist. Our proof uses a number theoretic classification of
approximations to algebraic numbers, together with a result of Lindenstrauss
and Weiss which is an application of Ratner's orbit closure theorem.
