In
algebraic number theory, the narrow class group of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
''K'' is a refinement of the
class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
of ''K'' that takes into account some information about embeddings of ''K'' into the field of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.
Formal definition
Suppose that ''K'' is a
finite extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...
of Q. Recall that the ordinary class group of ''K'' is defined to be
:
where ''I''
''K'' is the group of
fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral ...
s of ''K'', and ''P''
''K'' is the group 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
:
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
:
Uses
The narrow class group features prominently in the theory of representing of integers by
quadratic forms. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25).
:Theorem. Suppose that
::
:where ''d'' is a
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
, and that the narrow class group of ''K'' is trivial. Suppose that
::
:is a basis for the ring of integers of ''K''. Define a quadratic form
::
,
:where ''N''
''K''/Q is the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
. Then a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' is of the form
::
:for some integers ''x'' and ''y'' if and only if either
::
:or
::
:or
::
:where ''d''
''K'' is the
discriminant of ''K'', and
::
:indicates the
Legendre symbol.
Examples
For example, one can prove that the
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s Q(), Q(), Q() all have trivial narrow class group. Then, by choosing appropriate bases for the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s of each of these
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
s, the above theorem implies the following:
* A prime ''p'' is of the form ''p'' = ''x''
2 + ''y''
2 for integers ''x'' and ''y'' if and only if
::
: (This is known as
Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is true ar ...
.)
* A prime ''p'' is of the form ''p'' = ''x''
2 − 2''y''
2 for integers ''x'' and ''y'' if and only if
::
* A prime ''p'' is of the form ''p'' = ''x''
2 − ''xy'' + ''y''
2 for integers ''x'' and ''y'' if and only if
::
(cf.
Eisenstein prime
In mathematics, an Eisenstein prime is an Eisenstein integer
: z = a + b\,\omega, \quad \text \quad \omega = e^,
that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units , itself ...
)
An example that illustrates the difference between the narrow class group and the
usual class group is the case of Q(). 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'' = ''x''
2 - 6''y''
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'' = ''x''
2 - 6''y''
2 for integers ''x'' and ''y'' if and only if ''p'' = 3 or
::
(Whereas the first statement allows primes
, the second only allows primes
.)
See also
*
Class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
*
Quadratic form
References
* A. Fröhlich and M. J. Taylor, ''Algebraic Number Theory'' (p. 180), Cambridge University Press, 1991.
Algebraic number theory