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 , {\displaystyle 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 + , {\displaystyle 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 σ(a) is positive for every embedding
σ : K → R . {\displaystyle \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 = Q ( d ) , {\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
{ ω 1 , ω 2 } {\displaystyle \{\omega _{1},\omega _{2}\}\,\!}
is a basis for the ring of integers of K. Define a quadratic form
q K ( x , y ) = N K / Q ( ω 1 x + ω 2 y ) {\displaystyle 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 ) {\displaystyle p=q_{K}(x,y)\,}
for some integers x and y if and only if either
p ∣ d K , {\displaystyle p\mid d_{K}\,\!,}
or
p = 2 and d K ≡ 1 ( mod 8 ) , {\displaystyle p=2\quad {\mbox{ and }}\quad d_{K}\equiv 1{\pmod {8}},}
or
p > 2 and ( d K p ) = 1 , {\displaystyle p>2\quad {\mbox{ and}}\quad \left({\frac {d_{K}}{p}}\right)=1,}
where dK is the discriminant of K, and
( a b ) {\displaystyle \left({\frac {a}{b}}\right)}
denotes 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:
A prime p is of the form p = x2 + y2 for integers x and y if and only if
p = 2 or p ≡ 1 ( mod 4 ) . {\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
p = 2 or p ≡ 1 , 7 ( mod 8 ) . {\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
p = 3 or p ≡ 1 ( mod 3 ) . {\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:
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