  We solve models of $N$ species of fermions in the lowest Landau level with
$U(N)$-invariant interactions in the $N\gg 1$ limit. We find saddles of the
second quantized path integral at fixed chemical potential corresponding to
fractional Hall states with filling $ \frac{p}{q}$ where the integers $p$ and
$q$ depend on the chemical potential and interactions. On a long torus there
are $q$ such states related by translation symmetry, and $SU(N)$-invariant
excitations of fractional charge. Remarkably, these saddles and their filling
persist as extrema of the second-quantized action at $N=1$. Our construction
gives a first-principles derivation of fractional Hall states from strongly
interacting fermions.
