Narrow class group
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. == Formal definition == Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined as the quotient C K = I K / P K , {\displaystyle C_{K}=I_{K}/P_{K},\,} where IK is the group of fractional ideals of K, and PK is the subgroup of principal fractional ideals of K, that is, ideals of the form aOK where a is an element of K. The narrow class group is defined to be the quotient C K + = I K / P K + , {\displaystyle C_{K}^{+}=I_{K}/P_{K}^{+},} where now PK+ is the group of totally positive principal fractional ideals of K; that is, ideals of the form aOK where a is an element of K such that σ(a) is positive for every embedding σ : K → R .