Fabia water temperature warning light

On very cold days the red coolant / water temperature warning light and beeper comes on. The coolant level is fine and there are no leaks. This is a Skoda Fabia 1.4 mpi 8v car (2000). On warm days there is no warning light. Once on, the warning light stays on until the engine is turned off. How can I rectify this problem. Is it some sensor somewhere on the expansion tank? Also, as an unrelated problem the engine does not idle very well. Could this be another temperature sensor fault. I welcome any help given. Thank You.