Why does the radiator fan run even when the cars not started

If it's coming on even when the engine is cool you most likely have a relay issue to the fan or a control issue to the relay.

Check your temp sensor to see if it's working correctly

The fan on that vehicle is digitaly controlled by the vehicles DME if the fan is running all the time even with the vehicle off then te fan module (fan assembly itself) is faulty

I have a BMW e46 my fan runs non stop even when the car if off as a result my battery is dead, I had to disconnect the battrey but every time I connect the battrey it comes on again

I got the same issue this morning after installing a new stereo. I did not have to cut or splice in wires but once I reconnected the battery the fan came on even though the car was not on. Even when I start the car and try to adjust the fan it is not responding to the controls.