Nuclear-powered rockets were explored during the Cold War, and are a potential "thing," but there are significant contamination and safety issues, and high costs, and in the end did not prove sufficiently viable for the kinds of space travel (or weapons delivery) that we have done so far. But there are many who think that if we do engage in serious space-faring in the next century or so that nuclear-powered rockets may play a role in that because they have certain useful characteristics especially for long flights.

In the US, nuclear-powered rocket work began in 1958 as Project Rover, which eventually became NERVA (Nuclear Engine for Rocket Vehicle Application). They made significant progress but again, a nuclear-powered rocket never was judged as being necessary for our space usage thus far. The budgetary costs for the research were quite high and by the late 1970s the US had essentially stopped with the aggressive space program that had characterized its work through the Apollo missions. One can, in an indirect way, blame much of the dialing back of the US space program to the Vietnam War — its incredible cost meant that the US dialed back a lot of government funding projects, and it also soured the US public on space in general (which was widely seen as not addressing the important issues "on the ground" by the late 1960s, and was deeply connected to military activity). NERVA was cancelled in 1973, well before Three Mile Island or Chernobyl.

Other than using a nuclear reactor in a rocket body, you can also use nuclear explosions through even more ambitious programs like Project Orion. Some progress was made for Orion-style rockets but it was cancelled after the Limited Test Ban Treaty essentially made further development politically impossible (and it also would have great costs associated with it).

The issue is not that it is technically impossible. The issue is that human activities in space, combined with the political difficulties of high costs, have made this sort of thing very difficult. One could imagine a different context producing a different result. There would be down-sides to using nuclear rockets, but there are up- and down-sides to all technological choices.

The only nuclear technology presently in space are essentially "atomic batteries" (radioisotope thermoelectric generators) which convert the heat of nuclear decay into electricity for satellites. These produce very little power, but are useful for their very long lives.

If I were recommending one book on nuclear space stuff, it would probably be George Dyson's Project Orion. It is mostly about the Orion project, but it also discusses some of the other work.