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In this paper,we prove the existence of governing elds of the 2- rank and 4-rank of K2 (O)Q((∫)dp)as p varies.If d is divided by some prime p which is congruent to 3 or 5,then we prove that the governing eld of the 8-rank of K2 (O)Q((∫)dp)also exists.In other cases,we give some evidence to support the existence of the governing eld of the 8-rank of K2 (O)Q((∫)dp(2).