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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers. Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined to be C_K = I_K / P_K,,! where IK is the group of fractional ideals of K, and PK is the group of principal fractional ideals of K, that is, ideals of the form aOK where a is a unit of K. The narrow class group is defined to be the quotient C_K^+ = I_K / P_K^+.