you're right, they are completely different fields.
However one of the more likely methods of construction (from this distant point anyway) would be a launch of lots of material, tie it all to some asteroid or other large counterweight that has been put in space a good 90,000+km away, and then unspool it all to the surface, which would require a good and cheap heavy launch capacity
(I suppose you could say if we are going to do such an undertaking as build a space elevator, with all the billions it may entail, then launch costs to get it going will probably be the least of our worries, but as far as I see it, every little bit helps!)
The problem is not the principle of the thing, the problem is that a part of it would have to be made of unobtanium.
Earths gravity is such that you'd need a material for the tether that is stronger than any material that we are currently aware of.
So there is no point in getting any of the mechanisms in place or developed in any detail until there is a material that can withstand the forces in play. Right now, sad but true, an earthbound space elevator is science-fiction.
It kind of depends on which experts position paper you read on it - I point you in the direction of
10.1016/j.actaastro.2012.01.008 (Towards the Artsutanov's dream of the space elevator: The ultimate design of a 35 GPa strong tether thanks to graphene, N.M. Pugno)
For someone that is exploring the materials we will likely use in any attempt at an elevator.
Yes, we can't make the materials today. Like, right now. Or even next month, especially in the quantities needed. But we also can't make the rockets that might make it more accessible. And, more than likely, the first space elevator will be built on the moon, not on earth, which will definitely require heavy lift and would probably be a proving ground for the technology and materials.
So yes, science fiction, but so is pretty much everything else that spaceX wants to do.
They always say the future is just around the corner and in this case they might just be true. Or at least, as you can tell, I hope it to be true.
I thought maybe incorrectly that a carbon nanotube based material would provide the required strength? But we currently do not have the technology to build it?
The material doesn't exist, but the materials science certainly does. Carbon nanotubes are theoretically strong enough, and a ribbon made of nanotubes several centimeters long with a good epoxy gluing them together would do the trick, according to a NASA study by Bradley Edwards.
Nobody's achieved it yet, but plenty of people are working on it.
Rocketry has little to do with a space elevator, the materials science for a space elevator does not even exist.