Le groupe de Chow d'une surface de Ch{\^a}telet

Colliot-Th{\'e}l{\`e}ne has determined the Chow group of zero-cycles on a Ch{\^a}telet surface X defined over a finite extension K of the field of p-adic numbers (p an odd prime) when X is split by an unramified extension of K. Using similar techniques, we extend this calculation to the ramified case, simplifying his result in the process. We also determine how this group changes when we pass from K to a finite extension of K.