Earth's core may contain C, and it has been suggested that C in the core could stabilize the formation of a solid inner core composed of Fe3C. We experimentally examined the Fe-C system at a pressure of 5 GPa and determined the Fe-C phase diagram at this pressure. In addition, we measured solid metal/liquid metal partition coefficients for 17 trace elements and examined the partitioning behavior between Fe3C and liquid metal for 14 trace elements. Solid metal/liquid metal partition coefficients are similar to those found in one atmosphere studies, indicating that the effect of pressure to 5 GPa is negligible. All measured Fe3C/liquid metal partition coefficients investigated are less than one, such that all trace elements prefer the C-rich liquid to Fe3C. Fe3C/liquid metal partition coefficients tend to decrease with decreasing atomic radii within a given period. Of particular interest, our 5 GPa Fe-C phase diagram does not show any evidence that the Fe-Fe3C eutectic composition shifts to lower C contents with increasing pressure, which is central to the previous reasoning that the inner core may be composed of Fe3C.