84 olds omega

i have an 84 olds omega and i cant figure out if the alternator is gone or what. i have a fully charged battery i turn it over it fires up i let it run for 5 min then i shut it off i come back the next day to fire it up and wham it wont even turn over then i just get a dead battery so how do you test your alternator to see if it works and if it doesnt work properly could it drain the battery that easy?

Easiest thing to do anymore is take to the nearest Advance or Autozone and let them hook up testers to both battery and alternator. The way I check an alternator I would take a voltage meter ( a cheap one will do but get a digital one) and measure the voltage from the battery when the engine is off, and again when it is running. Be sure to have it set for proper voltage (DC). Should read 12+volts when off and 13-14+ volts when running. If the alternator is bad, the voltage will drop to 10-11 volts, or hover at or just below 12 volts. If it is not putting out more voltage than battery has when off, it is draining the battery, or weak enough that your lights and radio etc. would drain it.
Be sure to get the battery checked anyway. A failing battery will have voltage, but no strength. Only way to test that properly is to have a machine put a load on it and test it. May lose it's power after a few hours.

thats sounds all great and all but the battery is brand new im talking 1 week old so there cant be an issue.2nd how do you test the alternator when your car wont start up. ahaha im thinking its probably the starter now cause its definately a piece of .... but i thought maybe you could check cause im having issues even starting the car.