Narrow class group

Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined as the quotient

${\displaystyle C_{K}=I_{K}/P_{K},\,\!}$

where I_K is the group of fractional ideals of K, and P_K is the subgroup of principal fractional ideals of K, that is, ideals of the form aO_K where a is an element of K.

The narrow class group is defined to be the quotient

${\displaystyle C_{K}^{+}=I_{K}/P_{K}^{+},}$

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 an element of K such that σ(a) is positive for every embedding

${\displaystyle \sigma :K\to \mathbb {R} .}$

The narrow class group features prominently in the theory of representing integers by quadratic forms. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25).

Theorem. Suppose that ${\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,),}$ where d is a square-free integer, and that the narrow class group of K is trivial. Suppose that

${\displaystyle \{\omega _{1},\omega _{2}\}\,\!}$

is a basis for the ring of integers of K. Define a quadratic form

${\displaystyle q_{K}(x,y)=N_{K/\mathbb {Q} }(\omega _{1}x+\omega _{2}y)}$,

where N_K/Q is the norm. Then a prime number p is of the form

${\displaystyle p=q_{K}(x,y)\,\!}$

for some integers x and y if and only if either

${\displaystyle p\mid d_{K}\,\!,}$

or

${\displaystyle p=2\quad {\mbox{ and }}\quad d_{K}\equiv 1{\pmod {8}},}$

or

${\displaystyle p>2\quad {\mbox{ and}}\quad \left({\frac {d_{K}}{p}}\right)=1,}$

where d_K is the discriminant of K, and

${\displaystyle \left({\frac {a}{b}}\right)}$

denotes the Legendre symbol.

• Class group
• Quadratic form

• A. Fröhlich and M. J. Taylor, Algebraic Number Theory (p. 180), Cambridge University Press, 1991.