Narrow class group

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.

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:

• A prime p is of the form p = x² + y² for integers x and y if and only if

${\displaystyle p=2\quad {\mbox{or}}\quad p\equiv 1{\pmod {4}}.}$

(This is known as Fermat's theorem on sums of two squares.)

• A prime p is of the form p = x² − 2y² for integers x and y if and only if

${\displaystyle p=2\quad {\mbox{or}}\quad p\equiv 1,7{\pmod {8}}.}$

• A prime p is of the form p = x² − xy + y² for integers x and y if and only if

${\displaystyle p=3\quad {\mbox{or}}\quad p\equiv 1{\pmod {3}}.}$ (cf. Eisenstein prime)

An example that illustrates the difference between the narrow class group and the usual class group is the case of Q(√6). This has trivial class group, but its narrow class group has order 2. Because the class group is trivial, the following statement is true:

• A prime p or its inverse −p is of the form ± p = x² − 6y² for integers x and y if and only if

${\displaystyle p=2\quad {\mbox{or}}\quad p=3\quad {\mbox{or}}\quad \left({\frac {6}{p}}\right)=1.}$

However, this statement is false if we focus only on p and not −p (and is in fact even false for p = 2), because the narrow class group is nontrivial. The statement that classifies the positive p is the following:

• A prime p is of the form p = x² − 6y² for integers x and y if and only if p = 3 or

${\displaystyle \left({\frac {6}{p}}\right)=1\quad {\mbox{and}}\quad \left({\frac {-2}{p}}\right)=1.}$

(Whereas the first statement allows primes ${\displaystyle p\equiv 1,5,19,23{\pmod {24}}}$, the second only allows primes ${\displaystyle p\equiv 1,19{\pmod {24}}}$.)

• Class group
• Quadratic form

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