Is a Dhimmi put to death (Qisas) for killing a Muslim?

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A Dhimmi (protected non-Muslim subject) is executed for killing a Muslim. The evidence for this is the action of the Prophet, peace be upon him, who killed a Jew who had crushed the head of a young woman from the Ansar over jewelry (awdah). Furthermore, if retribution is exacted for killing with a lesser person, it is more appropriate when killing a superior person like a Muslim.