I have part of a tooth in my gum too (it sheared off at gum level years ago). I wouldn't let the dentist dig out the root. It is now a very small thing which has turned on it's side, obviously the tooth has been resorbed and only a small part of it is left. I could quite easily flick it out with my finger or toothbrush but am just leaving it for the time being as I am getting my remaining teeth extracted over the next few weeks. Our own bodies are quite adept at dealing with these things, even though it takes a long time. That particular tooth has never given me any problems like pain and I've never had any infections with it. When it finally falls out (unless it is removed first) I expect the gum to be completely healed underneath the same as the gum was when my front tooth fell out.