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
 * $$C_K = I_K / P_K,\,\!$$

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
 * $$C_K^+ = I_K / P_K^+,$$

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 &sigma;(a) is positive for every embedding
 * $$\sigma : K \to \mathbb{R}.$$

Uses
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 $$K = \mathbb{Q}(\sqrt{d}\,),$$ where d is a square-free 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
 * $$q_K(x,y) = N_{K/\mathbb{Q}}(\omega_1 x + \omega_2 y)$$,
 * where NK/Q is the 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
 * $$p > 2 \quad \mbox{ and} \quad \left(\frac {d_K} p\right) = 1,$$
 * where dK is the discriminant of K, and
 * $$\left(\frac ab\right)$$
 * denotes the Legendre symbol.

Examples
For example, one can prove that the quadratic fields Q($\sqrt{−1}$), Q($\sqrt{2}$), Q($\sqrt{−3}$) 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 + y&hairsp;2 for integers x and y if and only if
 * $$p = 2 \quad \mbox{or} \quad p \equiv 1 \pmod 4.$$
 * (This is known as Fermat's theorem on sums of two squares.)


 * A prime p is of the form p = x2 − 2y&hairsp;2 for integers x and y if and only if
 * $$p = 2 \quad \mbox{or} \quad p \equiv 1, 7 \pmod 8.$$


 * A prime p is of the form p = x2 − xy + y&hairsp;2 for integers x and y if and only if
 * $$p = 3 \quad \mbox{or} \quad p \equiv 1 \pmod 3.$$ (cf. Eisenstein prime)

An example that illustrates the difference between the narrow class group and the usual class group is the case of Q($\sqrt{6}$). 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 − 6y&hairsp;2 for integers x and y if and only if
 * $$p = 2 \quad \mbox{or} \quad p = 3 \quad \mbox{or} \quad \left(\frac{6}{p}\right)=1.$$

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 − 6y&hairsp;2 for integers x and y if and only if p = 3 or
 * $$\left(\frac{6}{p}\right)=1 \quad \mbox{and}\quad \left(\frac{-2}{p}\right)=1.$$

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