I went to an oral surgeon to have a tooth removed, The surgeon only removed part of the tooth, the root was left in and it is now infected.

Oral surgeons have a legal duty to comply with the standard of care. This means that they are required to do what other oral surgeons with similar training and experience would do under the same circumstances. If failing to remove the root during the first surgery was below the standard of care, then all of the costs associated with the second surgery should be the responsibiity of the oral surgeon, not yours.