“Will bone fragments dissolve after tooth extraction?”

No.

Yes they will

Bone fragments should dissolve by themselves if they are not larger than 3mm and unattached. For larger fragments, you might need to go back to the provider if they are causing you discomfort. Your body will slowly expel those fragments to the surface of the gum. The attached fragments might cause a feeling of "something pointy " on the gum. Those will dissolve by themselves as well. Painful fragments might need to be removed/smooth by your provider.