Three million is an absurd number, most put it somewhere around half a million. That aside, what you are asking gets into the idea of the "consumer city", which is a debate that is still pretty hot right now. Basically, a consumer city exists through the extraction of resources from the countryside without returning value (it gets a lot more complicated). It exists through systems of political power, and without those it would collapse. The early foundation of Alexandria most certainly fits this model, as it was planned out from the start to be a major capitol--but of course does not mean it remained a consumer city.

The other two types of cities are "producer cities" which survive through a relationship to the countryside (say, industrial manufacturing) that returns value and "merchant cities" which survive based on their relation to trade routes. It is worth noting that these are all "ideal types" and do not accurately reflect any complete historical reality, but they are useful for generally thinking about how cities operate.

Which one was Alexandria? It's debatable! But the important take away is that absolutely every city exists as a drain on rural resources because a large collection of people not producing food can only survive through food taken from those who are.

That's a novel, correct? Then be a little wary about taking what it says as fact. I don't mean that fiction can't be historically accurate, but I mean that by definition fiction isn't designed to be factual.

I have never seen any estimates of Alexandria's population anywhere near three million at the time of Caesar. I highly doubt it could be that high. Google says it is only four million now.

But large cities had to be supported by the countryside. Not only Alexandria, but Rome, Ephesus... any city you could name. A giant supply network had to be built to feed the inhabitants, and this was not always efficient. Just imagine all the manpower it would take, and all the equipment, not to mention losses and spoilage en route. So if you look at it like this, yes, Alexandria could have been a drain. Luckily Egypt was very fertile and could not only feed its own population but could even export food.

As for your question regarding the aristocracy and common Egyptian, I honestly don't know where to start. Is there something specific you were interested in?

Sorry to ask for a clarification.

I'm a huge fan of The First Man in Rome, but I don't remember Alexandria appearing in that novel, nor Caesar. The novel is set in the late second century before Caesar was born. Colleen McCullough is probably the best historical fiction author I know and I doubt she would make an error like this 3 million thing. Could you remind me where exactly Alexandria comes up so I can check to see what's going on?

I mean what was the relationship like between those in Alexandria of the ruling class and the rest of the classes in the city, and those in Egypt forced to supply them?

Edit: A word

I seriously doubt it. I'm currently studying Alexandria and Rome, and if anything, Alexandria had more than enough food for themselves, as well as sending hundreds of tons of grains out for Roman usage. Egypt was Rome's breadbasket.

Alexandria became a hotspot for all learning in the Roman Empire, with doctors from all over immigrating there to learn/practice medicine. The library of Alexandria gathered all the books they could for safekeeping there as well. Although there was some segregation between Greeks/Romans/Egyptians, the place was oftentimes considered the jewel of the world.