  We construct a $W_{\infty}$ gauge field theory of electrons in the lowest
Landau level. For this purpose we introduce an external gauge potential $\cal A
$ such that its $W_{\infty}$ gauge transformations cancel against the gauge
transformation of the electron field. We then show that the electromagnetic
interactions of electrons in the lowest Landau level are obtained through a
non-linear realization of $\cal A$ in terms of the $U(1)$ gauge potential
$A^{\m}$. As applications we derive the effective Lagrangians for circular
droplets and for the $\n =1$ quantum Hall system.
