Yes, indeed. Behold the HTRE-3 nuclear aircraft engine.
Yes, this is real. Yes, the military was crazy enough to think this was a good idea.
In principle, it’s straightforward. A jet engine is just a heat engine. You suck cold air in the front, you heat it up, you spew the hot air out the back. What we normally think of as a “jet engine” heats the air up by burning jet fuel. But the heat can come from anywhere. A nuclear reactor? No sweat. They get plenty hot.
Thing is, it’s easy to make a lightweight nuclear reactor that gets as hot as you want it to. The heavy part of a reactor isn’t the core, it’s the radiation shielding around the core.
So. What do you notice is missing in this picture?
Core’s looking a bit exposed, innit?
The US military built and flew these engines [edit: an experimental reactor similar to the one designed to drive these engines, not the engines themselves: see comments!] in a test aircraft.
They produce a lot of radiation, and you can’t easily shield them if you want them light enough to fly a plane. Which creates quite a problem for handling and servicing these things.
They considered a modular design where you could just undock the crew bits from the nasty radioactive bits…
…and then just reprocess it or, I don’t know, throw it away or something.
Then some engineer somewhere said “you guys do realize this idea is both crazy and stupid, right?” and the idea went away.