We study the dynamics of the charged and neutral excitons in a modulation-doped GaAs quantum well by time-resolved photoluminescence under a resonant excitation. The radiative lifetime of the charged exciton is found to be surprisingly short, 60 ps. This time is temperature independent between 2 and 10 K, and increases by a factor of 2 at 6 T. We discuss our findings in view of present theories of exciton radiative decay. [S0163-1829(98)03143-9].