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Algebraic number theory was extensively used in Wang’s joint research with Hua Loo-Keng in establishing the number-theoretic method for approximate calculation of integrals, which brought him back to the research on pure number theory for a period of time afterwards concluded with his book Diophantine Equations and Inequalities in Algebraic Number Fields. His findings in this regard have been celebrated among his foreign peers as ‘a valuable contribution to literature regarding Hardy-Littlewood Circle Method’. This part of Wang Yuan’s career is to be found in this chapter.