Is my alternator dead?

Use an inexpensive digital voltmeter to measure battery voltage while the battery light is on. It must be between 13.75 and 14.75 volts. If it is low, suspect worn brushes inside the alternator. They will cause an intermittent problem for a long time before they fail completely. The brush assembly can often be replaced without removing the alternator from the engine but most people just replace the entire alternator.

Could be a problem with the computer control circuit of the alternator's field generation

Next measure the voltage on the two small terminals on the back of the alternator, again, while the engine is running. They might be two wires that are plugged in or they could be two small nuts under a black plastic block that the wires go into. One will have full battery voltage. The other one is the key. It is supposed to be less than battery voltage but not 0 volts. If you find 0 volts, the brushes are open. If you find the same voltage on both terminals, either the voltage regulator circuit inside the Engine Computer is defective, (pretty rare), or the wire going to it is broken. If you find around 4 - 11 volts, that part of the system is working. There could be a defective diode in the alternator but I don't think that will turn the warning light on.