In mathematics, a ray class field is an
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
of a
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
associated with a ray class group of
ideal classes or
idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields.
The term "ray class group" is a translation of the German term "Strahlklassengruppe". Here "Strahl" is the German for a ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups. uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group.
There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated.
History
Weber introduced ray class groups in 1897. Takagi proved the existence of the corresponding ray class fields in about 1920. Chevalley reformulated the definition of ray class groups in terms of ideles in 1933.
Ray class fields using ideals
If ''m'' is an ideal of the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
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'' and ''S'' is a subset of the real places, then the ray class group of ''m'' and ''S'' is the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
:
where ''I''
''m'' 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 doma ...
s
co-prime to ''m'', and the "ray" ''P''
''m'' is the group of
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it ...
s generated by elements ''a'' with ''a'' ≡ 1 mod ''m'' that are positive at the places of ''S''.
When ''S'' consists of all real places, so that ''a'' is restricted to be totally positive, the group is called the narrow ray class group of ''m''. Some authors use the term "ray class group" to mean "narrow ray class group".
A ray class field of ''K'' is the abelian extension of ''K'' associated to a ray class group by class field theory, and its Galois group is isomorphic to the corresponding ray class group. The proof of existence of a ray class field of a given ray class group is long and indirect and there is in general no known easy way to construct it (though explicit constructions are known in some special cases such as imaginary quadratic fields).
Ray class fields using ideles
Chevalley redefined the ray class group of an ideal ''m'' and a set ''S'' of real places as the quotient of the idele class group by image of the group
:
where ''U''
''p'' is given by:
*The nonzero
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s for a complex place ''p''
*The positive
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 real ...
s for a real place ''p'' in ''S'', and all nonzero real numbers for ''p'' not in ''S''
*The units of ''K''
''p'' for a
finite place
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
''p'' not dividing ''m''
*The units of ''K''
''p'' congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to 1 mod ''p''
''n'' if ''p''
''n'' is the maximal power of ''p'' dividing ''m''.
Some authors use a more general definition, where the group ''U''
''p'' is allowed to be all nonzero real numbers for certain
real place
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
s ''p''.
The ray class groups defined using ideles are naturally isomorphic to those defined using ideals. They are sometimes easier to handle theoretically because they are all quotients of a single group, and thus easier to compare.
The ray class field of a ray class group is the (unique) abelian extension ''L'' of ''K'' such that the norm of the idele class group ''C''
''L'' of ''L'' is the image of
in the idele class group of ''K''.
Examples
If ''K'' is the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, ''m'' is a nonzero rational integer, and ''S'' comprises the
Archimedean place
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
of ''K'', then the ray class group of (''m'') and ''S'' is isomorphic to the group of units of Z/''m''Z, and the ray class field is the field generated by the ''m''th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. The ray class field for (''m'') and the empty set of places is its maximal totally real subfield -- the field
.
The
Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonicall ...
is the ray class field corresponding to the unit ideal and the empty set of real places, so it is the smallest ray class field. The
narrow Hilbert class field is the ray class field corresponding to the unit ideal and the set of all real places, so it is the smallest narrow ray class field.
References
*
*{{Neukirch ANT
Class field theory