How can you tell if a tooth is rotted?

Many ways. A "rotted" tooth often causes pain, or is sensitive to heat or cold. An x-ray can also be used to determine the health of a tooth. See your dentist as untreated tooth decay can lead to other medical problems.
Tooth discoloration. Tooth rotting when there has been so severe dental caries or trauma that the pulp of the tooth is dead. Signs and symptoms may include black or yellow discoloration of the enamel, indentation of the tooth inflammation or pus around the tooth and gums, severe pain, and sensitivity to temperature or certain foods. To prevent further infection it should be investigated preferably by a dentist.
See a dentist. My best recommendation is to see a dentist and have an examination done. Even if you were able to diagnose it properly and it was a cavity, you'd still need a dentist to look at it to determine what the proper treatment is for the problem.