What are the chances of getting an infection 23 days after wisdom teeth removal. and what would the symptoms be of an infection?

Slight: Not impossible, but not common. You may have trapped a foreign body in the socket (stitch, food debris,etc) and it is just now becoming a problem. More commonly, patients see a small often sharp piece coming through the tissue. This is a small fragment of loose bone, working its way out. No problem, no infection. It can be removed or come out on its own. See your DDS if needed. Good Luck.

Occasionally: you may develop a low-grade infection that occurs between three weeks to two months after wisdom tooth extraction. This occurs because either a piece of food or other debris got caught in the extraction site. See your surgeon for consultation.