  In this essay we explore the notion of essential dimension using the theory
of valuations of fields. Given a field extension K/k and a valuation on K that
is trivial on k, we prove that the rank of the valuation cannot exceed the
transcendence degree trdeg K. We use this inequality to prove lower bounds on
the essential dimension in some interesting situations. We study orbits of a
torus action and find a formula for the essential dimension of the functor of
these orbits.
