In a large pot with enough water to cover, soak the black-eyed peas 6 to 8 hours, or overnight.

Drain old water. Stir ham, carrot, onion, salt, and pepper into the pot. Cover all ingredients with fresh water, and bring to a boil. Reduce heat, and simmer 2 1/2 hours, or until black-eyed peas are tender. Stir occasionally and add water as necessary to keep ingredients covered.