We provide a uniform framework for proving the collapse of the hierarchy, NC1()for an exact arithmetic class of polynomial degree. These hierarchies collapses all the way down to the third level of the AC0-hierarchy, AC30(). Our main collapsing exhibits are the classes C=NC1C=LC=SAC1C=P NC1(C=L) and NC1(C=P) are already known to collapse \cite{ABO,Ogihara95,Ogiwara94}.

We reiterate that our contribution is a framework that works for \emph{all} these hierarchies.Our proof generalizes a proof from \cite{DMRTV} where it is used to prove thecollapse of the AC0(C=NC1) hierarchy. It is essentially based on a polynomialdegree characterization of each of the base classes.