  This article initiates the study of topological transcendental fields $\FF$
which are subfields of the topological field $\CC$ of all complex numbers such
that $\FF$ consists of only rational numbers and a nonempty set of
transcendental numbers. $\FF$, with the topology it inherits as a subspace of
$\CC$, is a topological field. Each topological transcendental field is a
separable metrizable zero-dimensional space and algebraically is $\QQ(T)$, the
extension of the field of rational numbers by a set $T$ of transcendental
numbers. It is proved that there exist precisely $2^{\aleph_0}$ countably
infinite topological transcendental fields and each is homeomorphic to the
space $\QQ$ of rational numbers with its usual topology. It is also shown that
there is a class of $2^{2^{\aleph_0} }$ of topological transcendental fields of
the form $\QQ(T)$ with $T$ a set of Liouville numbers, no two of which are
homeomorphic.
