A fault is (strongly) detectable if it is possible to construct a residual generator that is sensitive to the (constant) fault while decoupling all disturbances. Existing fault detectability criterions are reviewed and in two cases, improved versions are derived. For strong fault detectability, three new criterions are presented. To prove all criterions, a framework of polynomial bases is utilized. With these new and improved criterions, there exists now a criterion for models given both on transfer function form and state-space form, and for both fault detectability and strong fault detectability investigations. Recommendations are given on what criterion to use in different situations.