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. 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 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 an element of K. The narrow class group is defined to be the quotient 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 The narrow class group features prominently in the theory of representing of integers by quadratic forms. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25). For example, one can prove that the quadratic fields Q(√−1), Q(√2), Q(√−3) all have trivial narrow class group. Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following: