There is no way a nuclear rocket could take off from earth. Any method that might have the power to do it would irradiate a large area and a lot of the upper atmosphere. There are a few ways they could work in space:
[Blowing up bombs behind the vehicle](https://en.m.wikipedia.org/wiki/Nuclear_pulse_propulsion)
[Boiling the non nuclear fuel and squirting it out the back](https://en.m.wikipedia.org/wiki/Nuclear_thermal_rocket)
[Flinging very hot nuclear fuel out](https://en.m.wikipedia.org/wiki/Fission-fragment_rocket)
[To power an ion engine](https://en.m.wikipedia.org/wiki/Nuclear_electric_rocket)
In some cases nuclear decay is the method of generating the power/heat, rather than a nuclear reactor.
There’s more but I’m not sure what the differences are. As far as I know none have been tried in space, though there is radioactive decay generators powering some rovers/exploration craft. [There is other similar types of generator, but this sort of thing. ](https://en.m.wikipedia.org/wiki/Radioisotope_thermoelectric_generator)
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