2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150855We show that noncommutative differential forms on $k[x]$, $k$ a field, are of the form $Ω^1=k_λ[x]$ where $k_λ\supset k$ is a field extension. We compute the case $C\supset R$ explicitly, where $Ω^1$ is 2-dimensional. We study the induced quantum de Rahm complex, its cohomology and the associated moduli space of flat connections.Latex 19 pages no figures. Significant revision to give full moduli space of flat connections in Section 4Quantum Algebratext