Given the following context:  It was probably William the Conqueror who gave the city and its castle to Bishop Odo of Bayeux, the king's half brother. On William's death in September 1087 his territories were divided between his two sons. Robert, the elder, inherited the title of Duke of Normandy and William Rufus became King of England. A significant number of Norman barons objected to dividing Normandy and England, and Bishop Odo supported Robert's claim to the English throne. Several others, including the earls of Northumberland and Shrewsbury and the Bishop of Coutances came out in support of Robert. Odo prepared Rochester Castle for war and it became one of the headquarters of the rebellion. Its position in Kent made it a suitable base for raids on London and its garrison could harry William's forces in the county. William set off from London and marched towards Rochester to deal with the threat. Before he arrived, news reached the king that Odo had gone to Pevensey Castle, which was under the control of Robert, Count of Mortain. William turned away from Rochester and seized Pevensey. The captured Odo was forced to swear to hand over Rochester to William's men. The king despatched a force with Odo in tow to demand Rochester's surrender. Instead of yielding, the garrison sallied and captured the entire party. In response William laid siege to the city and castle. Contemporary chronicler Orderic Vitalis recorded that the siege began in May 1088. Two siege-castles were built to cut off the city's supply lines and to protect the besiegers from sorties. Conditions within the city were dire: disease was rampant, exacerbated by the heat and flies. The garrison ultimately capitulated and terms were agreed. Odo, Eustace, Count of Boulogne, and Robert de Belleme, son of the Earl of Shrewsbury, were allowed to march away with their weapons and horses but their estates in England were confiscated. This marked the end of the castle's role in the rebellion, and the fortification was probably abandoned shortly afterwards. The siege-castles were...  answer the following question:  What castle did William the Conqueror give to the king's half brother?
The answer to this question is:
Rochester Castle