Well, for one thing, you need to know your subject, and if you don’t even know what degrees are required to become a rocket scientist, then perhaps you shouldn’t write about one until you’ve done more research.
But the answer you’re looking for isn’t simple. There’s a course of study called “Aerospace Engineering,” but it’s a catch all, and many universities that offer it don’t necessarily offer advanced degrees in that subject that are called PhD’s. Where I went to school, that title was reserved for the Arts and Sciences.
But your rocket scientist could very well have a PhD in physics. Or planetary science. Or astrophysics. The painfully detailed specialized knowlege about rockets usually comes from work experience, not edumacation. So his work experience is at least as important as his schooling.
Some excellent places to have on your resume are Jet Propulsion Laboratory (atmospheric and planetary science missions), Goddard Space Flight Center (mission management), Rockwell (launch vehicles, now owned by…), Boeing (launch vehicles and manned space), Aerojet (now part of …) Northrop Grumman (another big name in manned space), Morton Thiokol (rocket engines), and The Aerospace Corporation (lots of consulting and spook work, employs many very smart people). Some very smart people have also worked for small upstart companies where they found the freedom to be creative and eccentric… Orbital Sciences Corp, Scaled Composites, Kistler Aerospace, Rotary Rocket, American Rocket… I’ve left out many other very plausible and important names in the space industry, but these are the ones folks will recognize.
One of the smartest people I know in the rocket science bizniz had a BS in electrical engineering from MIT, a Masters in Aerospace from Cal Tech, and a PhD in Physics from Cal Tech.
I suggest you do a bit more research into the career in general and in some detail if you pick a school or company for your character’s history. It helps immensely to know your subject well, and it can give you ideas for plot and characterization.