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This chapter presents the facilities provided in Magma for class field
theory.
The main objects of interest are abelian extensions of number fields
(FldAb) and maps between abelian groups and ideal groups.
Class field theory is concerned with the classification of all abelian
extensions of a given field. In particular, this covers abelian extensions
of number fields, local fields and global function fields. While
this chapter deals with the number field case only, Magma can
also perform computations in the other cases. For the case of global
function fields, see Chapter CLASS FIELD THEORY FOR GLOBAL FUNCTION FIELDS and for the case
of p-adic local fields, Section Class Field Theory.
Abstractly, class field theory parametrizes abelian extensions
in terms of abelian groups defined with respect to the base field.
In Magma, ray class groups and their quotients are used to define the
extensions. Ray class groups and their quotients are always represented as
maps between a finite abelian group (GrpAb) and the power structure
of ideals (PowIdeal). The maps are usually obtained as products
of the map returned by ray class groups and quotient maps.
In theory, a class field is completely determined by the corresponding
class group (map). Currently there is only a small number of invariants
that can be
computed directly from the map; for most other properties the class
field has to be converted into a number field by computing a set of
defining equations.
Class field theory classifies all abelian extensions of a given number field
k in terms of quotients of ray class groups. Ray class groups should be
thought of as generalized class groups in that
they can be defined similarly to class groups:
1 to U to k * to I to Cl to 1
where k * is the multiplicative group of k, U is the group of units,
and I the group of fractional ideals.
(Recall, the class group is defined as the quotient of the ideals modulo the
principal ideals).
To define ray class groups we need to refine all of the above terms.
Let o = Zk be the ring of integers in k and fix an integral ideal m0
in o. Furthermore, let m_∞ be a subset of the real embeddings of k
into C and, formally, m := (m0, m_∞). Then for x ∈k define
x mod * m = 1 iff Biggl{
((vp(x - 1) ≥vp(m0) for all prime ideals p)
atop
(s(x)>0 for s ∈m_∞))
Let Im denote the group of fractional ideals that are coprime to m0,
Um the units u such that u mod * m = 1 and
Pm the elements x∈k such that x mod * m = 1. Then
1 to Um to Pm to Im to Clmto 1
defines the ray class groups modulo m.
For m = (1o, emptyset), this corresponds to the usual class group.
Given two moduli m and n such that m0|n0 and
m_∞⊆n_∞ we have canonical surjections
Cln to Clm.
Now, let H be a subgroup such that Pm < H < Im.
There exists a minimal n such that
the canonical embedding Clm/H to Cln is injective. This n is called
the conductor of H.
The main theorem of class field theory asserts the following
correspondence between
abelian extensions K/k of k and quotients of ray class groups:
Let K/k be abelian and G := Gal(K/k) the group of k-automorphisms of K.
For each prime ideal p of k that is unramified in K, let
Frob(p, K/k) be the Frobenius automorphism, i.e. the unique s∈G
such that s(x) = xN mod p for all x∈k.
Extend this map multiplicatively to all ideals coprime to dK/k, the
discriminant of K/k.
This is known as the Artin-map and is denoted by (a, K/k).
Class field theory asserts the existence
of some modulus m and a subgroup H such that
G isomorphic to Clm/H by the Artin-map.
Conversely, for each ray class group Clm and each subgroup Pm< H there
is an abelian extension K/k with the above property.
The correspondence is one-to-one if one restricts to pairs
Clm and H such that m is the conductor of H.
For m = (1o, emptyset), the corresponding field K is called the
Hilbert class field of k (in the wide sense).
Perhaps best understood is the Hilbert class field. As conjectured
by Hilbert and proved by Furtwängler, all ideals of k become
principal in the Hilbert class field Hk. Furthermore, Hk is the
maximal unramified abelian extension of k.
A different interpretation of H is provided by
norm groups. We have
H = < NK/k(a) | a ∈IKm >
This result may be used to convert an abelian number field into an
abelian extension. In addition, class field theory asserts that,
for an arbitrary normal extension K/k, the norm group defined above
corresponds to the maximal abelian subfield.
In Magma maps are used to represent the ideal groups. If necessary, the map
can be augmented by supplying m i.e. an integral ideal m0 and a sequence
of indices of the real places in m_∞. The map returned as a
second return value from ClassGroup or
RayClassGroup
carries the necessary information to recover m, even if this
is hidden from the user. As an example, we consider the Hilbert class field
of k := Q(α) with α3 + α2 + 3α - 6 = 0
with class group C4.
> k := NumberField(Polynomial([ -6, 3, 1, 1 ]));
Now, the easiest way to get the Hilbert class field is to call
HilbertClassField directly on k:
> K := HilbertClassField(k);
> K;
Number Field with defining polynomial $.1^4 + (76*k.1^2 - 420*k.1 - 488)*$.1^2\
+ 32080*k.1^2 + 41984*k.1 + 95168 over k
Let us now verify some of the properties, starting with the discriminant.
Since K/k is totally unramified, the discriminant of the maximal order should
be 1:
> O := MaximalOrder(K);
> O;
Maximal Order of Equation Order with defining polynomial x^4 + [-488, -420,
76]*x^2 + [95168, 41984, 32080] over its ground order
> Discriminant(O);
Ideal
Basis:
[1 0 0]
[0 1 0]
[0 0 1]
Now let us check that all ideals of k are principal in K.
In order to do so, it is sufficient to demonstrate that a generator
of the class group becomes principal:
> g, m := ClassGroup(k);
> g;m;
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*g.1 = 0
Mapping from: GrpAb: g to Set of ideals of Maximal Equation Order with
defining polynomial x^3 + x^2 + 3*x - 6 over its ground order
> i := m(g.1);
> I := O!!i;
> IsPrincipal(I);
true
> f,g := IsPrincipal(I);
> g;
[[2193497788678474035456, -1238066307883451022336,
-2319265120953032748288], [394272965034846395136,
-222653406238254306432, -417025104282566694144],
[167087257584, 71708840496, 32825469008], [495632919,
-279332235, -523526213]] / 10917386545536
However, to use the more sophisticated functions, the construction of the
class fields needs to be done step--by--step.
We first create an abelian extension as an object of type FldAb.
Since we wish to see how much of the class group becomes trivial
in the quadratic subfield K 1 of K (capitulates in K 1),
we also define this.
> aK := AbelianExtension(m);
> g, m := ClassGroup(k);
> q, mq := quo<g | 2*g.1>;
> m2 := Inverse(mq)*m;m2;
Mapping from: GrpAb: q to Set of ideals of Maximal
Equation Order with defining polynomial x^3 + x^2 + 3*x
- 6 over Z
Composition of Mapping from: GrpAb: q to GrpAb: g and
Mapping from: GrpAb: g to Set of ideals of Maximal
Equation Order with defining polynomial x^3 + x^2 + 3*x
- 6 over Z
> aK2 := AbelianExtension(m2);
This demonstrates a very important technique: the creation of a
quotient group of the class group together with the corresponding
map.
A few invariants such as degree and discriminant may be calculated
directly from
aK without the (costly) computation of a defining equation.
> Discriminant(aK);
Principal Ideal
Generator:
[1, 0, 0]
[ 4, 4 ]
The second return value denotes the signature of the field
as an extension of Q.
Now we compute a defining equation for aK2 and see what happens
to the class group of k.
> O := MaximalOrder(aK2);O;
Maximal Order of Equation Order with defining
polynomial x^2 + [-5, -2, -1] over its ground order
> O!!m(g.1);
> IsPrincipal($1);
false
> IsPrincipal($2^2);
true
So only "half" of the class group capitulates in aK2, the
remaining part collapses in aK.
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