Battery light illuminated means the car is running off the battery, not the alternator. If you jump start it with a dead battery and an alternator issue the car would cut out not long after jump starting. If you're having starting issues and the battery light is on it's because the battery isn't getting charged properly by the alternator. If the battery goes too low too often then you may need to replace the battery too if the alternator is found to be the fault.
Have you checked the alternator belt and the wires attached to it?
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