In
algebraic number theory
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 ...
, the Hilbert class field ''E'' 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 the
maximal abelian unramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of ''E'' over ''K'' is canonically isomorphic to the
ideal 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'' using
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
s for
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s in ''K''.
In this context, the Hilbert class field of ''K'' is not just unramified at the
finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every
real embedding
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfie ...
of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E'').
Examples
*If the ring of integers of ''K'' is a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
, in particular if
, then ''K'' is its own Hilbert class field.
*Let
of
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
. The field
has discriminant
and so is an everywhere unramified extension of ''K'', and it is abelian. Using the
Minkowski bound
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number field ''K''. It is named for the mathematician Hermann Minkowski.
Definition
Let ''D'' be th ...
, one can show that ''K'' has class number 2. Hence, its Hilbert class field is
. A non-principal ideal of ''K'' is (2,(1+)/2), and in ''L'' this becomes the principal ideal ((1+)/2).
*The field
has class number 3. Its Hilbert class field can be formed by adjoining a root of x
3 - x - 1, which has discriminant -23.
*To see why ramification at the archimedean primes must be taken into account, consider the
real
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)
...
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 an ...
''K'' obtained by adjoining the square root of 3 to Q. This field has class number 1 and discriminant 12, but the extension ''K''(''i'')/''K'' of discriminant 9=3
2 is unramified at all prime ideals in ''K'', so ''K'' admits finite abelian extensions of degree greater than 1 in which all finite primes of ''K'' are unramified. This doesn't contradict the Hilbert class field of ''K'' being ''K'' itself: every proper finite abelian extension of ''K'' must ramify at some place, and in the extension ''K''(''i'')/''K'' there is ramification at the archimedean places: the real embeddings of ''K'' extend to complex (rather than real) embeddings of ''K''(''i'').
*By the theory of
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
, the Hilbert class field of an
imaginary 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 an ...
is generated by the value of the
elliptic modular function
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...
at a generator for the ring of integers (as a Z-module).
History
The existence of a (narrow) Hilbert class field for a given number field ''K'' was conjectured by and proved by
Philipp Furtwängler
Friederich Pius Philipp Furtwängler (April 21, 1869 – May 19, 1940) was a German number theorist.
Biography
Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (''Zur Theorie der in Linearfaktoren ze ...
. The existence of the Hilbert class field is a valuable tool in studying the structure of the
ideal 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 a given field.
Additional properties
The Hilbert class field ''E'' also satisfies the following:
*''E'' is a finite Galois
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
of ''K'' and
'E'' :'' K''= ''h''
''K'', where ''h''
''K'' is the
class number of ''K''.
*The
ideal 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'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of ''E'' over ''K''.
*Every
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of ''O''
''K'' extends to a
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 ...
of the ring extension ''O''
''E'' (
principal ideal theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, whic ...
).
*Every
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
''P'' of ''O''
''K'' decomposes into the product of ''h''
''K'' / ''f'' prime ideals in ''O''
''E'', where ''f'' is the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of
'P''in the ideal class group of ''O''
''K''.
In fact, ''E'' is the unique
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 ...
satisfying the first, second, and fourth properties.
Explicit constructions
If ''K'' is imaginary quadratic and ''A'' is an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
with
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
by 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 ''K'', then adjoining the
j-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such funct ...
of ''A'' to ''K'' gives the Hilbert class field.
[Theorem II.4.1 of ]
Generalizations
In
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
, one studies the
ray class field In mathematics, a ray class field is an abelian extension of a global field 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 te ...
with respect to a given
modulus, which is a formal product of prime ideals (including, possibly, archimedean ones). The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. The Hilbert class field is then the ray class field with respect to the trivial modulus ''1''.
The ''narrow class field'' is the ray class field with respect to the modulus consisting of all infinite primes. For example, the argument above shows that
is the narrow class field of
.
Notes
References
*
*
*
* J. S. Milne, Class Field Theory (Course notes available at http://www.jmilne.org/math/). See the Introduction chapter of the notes, especially p. 4.
*
*
{{PlanetMath attribution, id=2870, title=Existence of Hilbert class field
Class field theory