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Module is generalization vector space which algebraic properties does not hold.  The module has a basis is called free module. In this research, we investigated M module over  Quadratic Complex Field Q[√D]. we showed  cardinality of basis module 〖free-rank 〗_Q (M)≤〖rank〗_Q  (M). We give example higher order D=167,283, and 11283 to count cardinality of basis M. Let M be a module over quadratic field, any free module is torsion module but not free-torsion module. Because quadratic complex field is unique factorization domain, thus every submodule N of  M over quadratic field has minimal cardinal a basis which is .〖free-rank 〗_Q (M)≤〖free-rank〗_Q  (N).

Free Module over Quadratic Field, Torsion Module, Unique Factorization Domain