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
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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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
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- is a basis for the ring of integers of K. Define a quadratic form
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = q_K(x,y)\,\!}
- for some integers x and y if and only if either
- or
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = 2 \quad \mbox{ and } \quad d_K \equiv 1 \pmod 8,}
- or
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p > 2 \quad \mbox{ and} \quad \left(\frac {d_K} p\right) = 1,}
- where dK is the discriminant of K, and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac ab\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 + y 2 for integers x and y if and only if
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 − 2y 2 for integers x and y if and only if
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = 2 \quad \mbox{or} \quad p \equiv 1, 7 \pmod 8.}
- A prime p is of the form p = x2 − xy + y 2 for integers x and y if and only if
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 − 6y 2 for integers x and y if and only if
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 2 for integers x and y if and only if p = 3 or
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{6}{p}\right)=1 \quad \mbox{and}\quad \left(\frac{-2}{p}\right)=1.}
(Whereas the first statement allows primes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \equiv 1, 5, 19, 23 \pmod {24}} , the second only allows primes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \equiv 1, 19 \pmod {24}} .)
See also
References
- A. Fröhlich and M. J. Taylor, Algebraic Number Theory (p. 180), Cambridge University Press, 1991.