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.
= Motivation =
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).
:Theorem. Suppose that ::K = mathbf Q(sqrt d), :where d is a squarefree integer, and that the narrow class group of K is trivial. Suppose that ::{ omega_1, omega_2 },! :is a basis for the ring of integers of K . Define a quadratic form :: q_K(x,y) = N_{K/mathbf Q}(omega_1 x + omega_2 y), :where N K /Q is the field norm. Then a prime number p is of the form :: p = q_K(x,y),! :for some integers x and y if and only if either :: p mid d_K,!, :or :: p = 2 quad mbox{and} quad d_K equiv 1 pmod 8, :or :: p > 2 quad mbox{and} quad left(frac {d_K} p ight) = 1, :where d K is the Discriminant#Discriminant of an algebraic number field of K , and ::left(frac ab ight) :indicates the Legendre symbol.
== Examples ==
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:
= Formal definition =
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 I K is the group of fractional ideals of K , and P K is the group of principal fractional ideals of K , that is, ideals of the form aO K where a is a Unit (ring theory) of K .
The narrow class group is defined to be the quotient :C_K^+ = I_K / P_K^+, where now P K + is the group of totally positive principal fractional ideals of K ; that is, ideals of the form aO K where a is a unit of K such that σ( a ) is positive for every embedding :sigma : K o mathbf R.
= See also =
= References =
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