  We investigate fields of characteristic 0 and dp-rank 2. While we do not
obtain a classification, we prove that any unstable field of characteristic 0
and dp-rank 2 admits a unique definable V-topology. If this statement could be
generalized to higher ranks, we would obtain the expected classification of
fields of finite dp-rank.
  We obtain the unique definable V-topology by investigating the "canonical
topology" defined in earlier work. Contrary to earlier expectations, the
canonical topology need not be a V-topology. However, we are able to
characterize the canonical topology (on fields of dp-rank 2 and characteristic
0) in terms of differential valued fields.
  This differential valued structure is obtained through a partial
classification of "2-inflators," a sort of generalized valuation that arises
naturally in fields of finite rank.
  Additionally, we give an example of a dp-rank 2 expansion of ACVF with a
definable set of full rank and empty interior. This example interferes with
certain strategies for proving the henselianity conjecture.
