In　this　paper,　we　investigate　the　Rayleigh-Taylor　instability　problem　for　two　compressible,　immiscible,　inviscid　flows　rotating　with　a　constant　angular　velocity,　and　evolving　with　a　free　interface　in　the　presence　of　a　uniform　gravitational　field.　First　we　construct　the　Rayleigh-Taylor　steady-state　solutions　with　a　denser　fluid　lying　above　the　free　interface　with　the　second　fluid,　then　we　turn　to　an　analysis　of　the　equations　obtained　from　linearization　around　such　a　steady　state.　In　the　presence　of　uniform　rotation,　there　is　no　natural　variational　framework　for　constructing　growing　mode　solutions　to　the　linearized　problem.　Using　the　general　method　of　studying　a　family　of　modified　variational　problems　introduced　in　[1],　we　construct　normal　mode　solutions　that　grow　exponentially　in　time　with　rate　like,　where　ζ　is　the　spatial　frequency　of　the　normal　mode　and　the　constant　c　depends　on　some　physical　parameters　of　the　two　layer　fluids.　A　Fourier　synthesis　of　these　normal　mode　solutions　allows　us　to　construct　solutions　that　grow　arbitrarily　quickly　in　the　Sobolev　space　Hk,　and　leads　to　an　ill-posedness　result　for　the　linearized　problem.　Moreover,　from　the　analysis　we　see　that　rotation　diminishes　the　growth　of　instability.　Using　the　pathological　solutions,　we　then　demonstrate　the　ill-posedness　for　the　original　non-linear　problem　in　some　sense.　©　2015　Wuhan　Institute　of　Physics　and　Mathematics.