[snrplfth]

Humans already have the technology - nuclear explosives, and a lot of them.

And I don't see how it's a bad time for space-oriented stuff in the USA - it hasn't been this good for years! What makes you think NASA is going to get defunded?

[astrodust]

Nuclear weapons aren't especially useful for pushing things around. You need a sustained, directed force to make any meaningful change in trajectory.

Nuking an asteroid would likely just bust it up into smaller chunks that are still going in roughly the same direction. Instead of one threat you now have millions.

[snrplfth]

The usefulness of nuclear explosives is in their high energy-to-weight ratio. It's very easy to mount them on rockets, guide them to targets and detonate them precisely. A short burst is as good as a long push if the total force is the same, and nuclear explosives provide lots of that.

Most ideas for using nuclear explosives for asteroid redirection don't involve buryingit into the surface, but rather a "stand-off" explosion that occurs some distance from the asteroid's surface. This would massively heat that side of the asteroid and cause a layer to puff off, acting as reaction mass, and not fracturing the asteroid itself.

And if an asteroid really was blown apart into smaller fragments, that would be an improvement. Even a small relative velocity between fragments would spread them out to a much larger area than the earth's diameter, meaning that fewer would probably hit. Also, smaller meteorites are typically much less destructive than larger ones, measured by weight. It's not linear - one hundred meteors of a hundred tons each is basically harmless, but one ten thousand ton meteor can blow a chunk out of a city. Blowing an asteroid into a 'million fragments' would be a job well done.