Oral Surgery. Go see a dentist that has the qualifications to remove the broken root. This can be a difficult surgery. I would recommend treating it soon before you get an infection. I would also recommend doing a graft after the root is removed which will help preserve the remaining bone.
Surgical extraction. When a tooth is fractured at the gumline, there is nothing for a dentist to hold on to remove it. Very common after a root canal because the tooth becomes more brittle after the removal of its' nerve. To extract this tooth, we must take a drill and section the tooth into pieces.This gives us a purchase point to hold the pieces and then remove them. Healing is just as quick as a regular extraction.