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

:$C\_K\; =\; I\_K\; /\; P\_K,\backslash ,\backslash !$

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

:$C\_K^+\; =\; I\_K\; /\; P\_K^+,$

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

:$\backslash sigma\; :\; K\; \backslash to\; \backslash mathbb\{R\}.$

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

::$\backslash \{\; \backslash omega\_1,\; \backslash omega\_2\; \backslash \}\backslash ,\backslash !$

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

:: $q\_K(x,y)\; =\; N\_\{K/\backslash mathbb\{Q\}\}(\backslash omega\_1\; x\; +\; \backslash omega\_2\; y)$,

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

:: $p\; =\; q\_K(x,y)\backslash ,\backslash !$

:for some integers x and y if and only if either

:: $p\; \backslash mid\; d\_K\backslash ,\backslash !,$

:or

:: $p\; =\; 2\; \backslash quad\; \backslash mbox\{\; and\; \}\; \backslash quad\; d\_K\; \backslash equiv\; 1\; \backslash pmod\; 8,$

:or

:: $p\; >\; 2\; \backslash quad\; \backslash mbox\{\; and\}\; \backslash quad\; \backslash left(\backslash frac\; \{d\_K\}\; p\backslash right)\; =\; 1,$

:where d_K is the discriminant of K, and

::$\backslash left(\backslash frac\; ab\backslash right)$

:denotes the Legendre symbol.

For example, one can prove that the quadratic fields Q({{radic|−1}}), Q({{radic|2}}), Q({{radic|−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

::$p\; =\; 2\; \backslash quad\; \backslash mbox\{or\}\; \backslash quad\; p\; \backslash equiv\; 1\; \backslash pmod\; 4.$

: (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

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

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

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

An example that illustrates the difference between the narrow class group and the usual class group is the case of Q({{radic|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

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

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

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

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

• Class group
• Quadratic form

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

Category:Algebraic number theory