Can you blow up a nuke in space?

It's possible, though it's (probably... see below) prohibited by both the Outer Space Treaty and the Partial Test Ban Treaty of 1963. All countries known to possess nuclear weapons capability, with the exception of North Korea, have ratified at least one of these (North Korea has signed, but not ratified, the Outer Space Treaty).

Whether or not a non-weapon nuclear explosion (for example: using a nuclear device as a sort of improved dynamite for, say, mining purposes) would violate the Outer Space Treaty is an interesting legal question, since the treaty expressly refers to nuclear weapons and goes on to say that the use of any equipment or facility necessary for peaceful exploration shall not be prohibited. This could potentially eventually become important.