  We present field theoretical descriptions of massless (2+1) dimensional
nonrelativistic fermions in an external magnetic field, in terms of a fermionic
and bosonic second quantized language. An infinite dimensional algebra,
$W_{\infty}$, appears as the algebra of unitary transformations which preserve
the lowest Landau level condition and the particle number. In the droplet
approximation it reduces to the algebra of area-preserving diffeomorphisms,
which is responsible for the existence of a universal chiral boson Lagrangian
independent of the electrostatic potential. We argue that the bosonic droplet
approximation is the strong magnetic field limit of the fermionic theory. The
relation to the $c=1$ string model is discussed.
