Does the third field stay forever once you have 3 coins to buy it? If not when does it disappear? The rules aren't terribly clear (or maybe I should say "are terribly clear").
You keep the third bean field for the duration of the game. It would be harsh if it was limited as it does cost you 3 gold coins (victory points).
Here is a discussion about when to buy or not buy that third field. Due to the cost, it is not always the right thing to do to purchase the field.