# Class Field Theory

## Introduction

This chapter deals with abelian extensions of number fields and the rational numbers.

Class Field Theory, here specifically, class field theory of global number fields, deals with abelian extension, ie. fields where the group of automorphisms is abelian. For extensions of $\mathbb Q$, the famous Kronnecker-Weber theorem classifies all such fields: a field is abelian if and only if it is contained in some cyclotomic field. For general number fields this is more involved and even for extensions of $\mathbb Q$ is is not practical.

In Hecke, abelian extensions are parametrized by quotients of so called ray class groups. The language of ray class groups while dated is more applicable to algorithms than the modern language of idel class groups and quotients.

## Ray Class Groups

Given an integral ideal $m_0 \le Z_K$ and a list of real places $m_\infty$, the ray class group modulo $(m_0, m_\infty)$, $C(m)$ is defined as the group of ideals coprime to $m_0$ modulo the elements $a\in K^*$ s.th. $v_p(a-1) \ge v_p(m_0)$ and for all $v\in m_\infty$, $a^{(v)} >0$. This is a finite abelian group. For $m_0 = Z_K$ and $m_\infty = \{\}$ we get $C()$ is the class group, if $m_\infty$ contains all real places, we obtain the narrow class group, or strict class group.

`ray_class_group`

— Method`ray_class_group(m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}; n_quo::Int, lp::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}) -> FinGenAbGroup, MapRayClassGrp`

Given an ideal $m$ and a set of infinite places of $K$, this function returns the corresponding ray class group as an abstract group $\mathcal {Cl}_m$ and a map going from the group into the group of ideals of $K$ that are coprime to $m$. If `n_quo`

is set, it will return the group modulo `n_quo`

. The factorization of $m$ can be given with the keyword argument `lp`

.

`class_group`

— Method`class_group(K::AbsSimpleNumField) -> FinGenAbGroup, Map`

Shortcut for `class_group(maximal_order(K))`

: returns the class group as an abelian group and a map from this group to the set of ideals of the maximal order.

`norm_group`

— Method```
norm_group(f::Nemo.PolyRingElem, mR::Hecke.MapRayClassGrp, is_abelian::Bool = true; of_closure::Bool = false) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
norm_group(f::Array{PolyRingElem{AbsSimpleNumFieldElem}}, mR::Hecke.MapRayClassGrp, is_abelian::Bool = true; of_closure::Bool = false) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
```

Computes the subgroup of the Ray Class Group $R$ given by the norm of the extension generated by a/the roots of $f$. If `is_abelian`

is set to true, then the code assumes the field to be abelian, hence the algorithm stops when the quotient by the norm group has the correct order. Even though the algorithm is probabilistic by nature, in this case the result is guaranteed. If `of_closure`

is given, then the norm group of the splitting field of the polynomial(s) is computed. It is the callers responsibility to ensure that the ray class group passed in is large enough.

`norm_group`

— Method```
norm_group(K::RelSimpleNumField{AbsSimpleNumFieldElem}, mR::Hecke.MapRayClassGrp) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
norm_group(K::RelNonSimpleNumField{AbsSimpleNumFieldElem}, mR::Hecke.MapRayClassGrp) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
```

Computes the subgroup of the Ray Class Group $R$ given by the norm of the extension.

## Ray Class Fields

In general, the construction of a class field starts with a (ray) class group. Each quotient of a ray class group then defines a ray class field, the defining property is that the (relative) automorphism group is canonically isomorphic to the quotient of the ray class group where the isomorphism is given by the Artin (or Frobenius) map. Since, in Hecke, the (ray) class groups have no link to the field, actually this has to be specified using the maps.

It should be noted that this is a *lazy* construction: nothing is computed at this point.

`ray_class_field`

— Method```
ray_class_field(m::MapClassGrp) -> ClassField
ray_class_field(m::MapRayClassGrp) -> ClassField
```

Creates the (formal) abelian extension defined by the map $m: A \to I$ where $I$ is the set of ideals coprime to the modulus defining $m$ and $A$ is a quotient of the ray class group (or class group). The map $m$ must be the map returned from a call to {class*group} or {ray*class_group}.

`ray_class_field`

— Method`ray_class_field(m::Union{MapClassGrp, MapRayClassGrp}, quomap::FinGenAbGroupHom) -> ClassField`

For $m$ a map computed by either {ray*class*group} or {class_group} and $q$ a canonical projection (quotient map) as returned by {quo} for q quotient of the domain of $m$ and a subgroup of $m$, create the (formal) abelian extension where the (relative) automorphism group is canonically isomorphic to the codomain of $q$.

`ray_class_field`

— Method`ray_class_field(I::AbsNumFieldOrderIdeal; n_quo = 0) -> ClassField`

The ray class field modulo $I$. If `n_quo`

is given, then the largest subfield of exponent $n$ is computed.

`ray_class_field`

— Method`ray_class_field(I::AbsNumFieldOrderIdeal, inf::Vector{InfPlc}; n_quo = 0) -> ClassField`

The ray class field modulo $I$ and the infinite places given. If `n_quo`

is given, then the largest subfield of exponent $n$ is computed.

`hilbert_class_field`

— Method`hilbert_class_field(k::AbsSimpleNumField) -> ClassField`

The Hilbert class field of $k$ as a formal (ray-) class field.

`ring_class_field`

— Method`ring_class_field(O::AbsNumFieldOrder) -> ClassField`

The ring class field of $O$, i.e. the maximal abelian extension ramified only at primes dividing the conductor with the automorphism group isomorphic to the Picard group.

### Example

`julia> Qx, x = polynomial_ring(FlintQQ, "x");`

`julia> K, a = number_field(x^2 - 10, "a");`

`julia> c, mc = class_group(K)`

`(Z/2, ClassGroup map of Set of ideals of Maximal order of Number field of degree 2 over QQ with basis AbsSimpleNumFieldElem[1, a])`

`julia> A = ray_class_field(mc)`

`Class field defined mod (<1, 1>, InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}[]) of structure Z/2`

## Conversions

Given a ray class field, it is possible to actually compute defining equation(s) for this field. In general, the number field constructed this way will be non-simple by type and is defined by a polynomial for each maximal cyclic quotient of prime power order in the defining group.

The algorithm employed is based on Kummer-theory and requires the addition of a suitable root of unity. Progress can be monitored by setting `set_verbose_level(:ClassField, n)`

where $0\le n\le 3$

`number_field`

— Method`number_field(CF::ClassField) -> RelNonSimpleNumField{AbsSimpleNumFieldElem}`

Given a (formal) abelian extension, compute the class field by finding defining polynomials for all prime power cyclic subfields.

Note, the return type is always a non-simple extension.

`julia> Qx, x = polynomial_ring(FlintQQ, "x");`

`julia> k, a = number_field(x^2 - 10, "a");`

`julia> c, mc = class_group(k);`

`julia> A = ray_class_field(mc)`

`Class field defined mod (<1, 1>, InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}[]) of structure Z/2`

`julia> K = number_field(A)`

`Non-simple number field with defining polynomials [x^2 - 2] over number field with defining polynomial x^2 - 10 over rational field`

`julia> ZK = maximal_order(K)`

`Relative maximal order of Non-simple number field of degree 2 over number field with pseudo-basis (1, 1//1 * <1, 1>) (_$1 + a, 1//4 * <2, a>)`

`julia> isone(discriminant(ZK))`

`true`

`ray_class_field`

— Method```
ray_class_field(K::RelSimpleNumField{AbsSimpleNumFieldElem}) -> ClassField
ray_class_field(K::AbsSimpleNumField) -> ClassField
```

For a (relative) abelian extension, compute an abstract representation as a class field.

`genus_field`

— Method`genus_field(A::ClassField, k::AbsSimpleNumField) -> ClassField`

The maximal extension contained in $A$ that is the compositum of $K$ with an abelian extension of $k$.

`maximal_abelian_subfield`

— Method`maximal_abelian_subfield(A::ClassField, k::AbsSimpleNumField) -> ClassField`

The maximal abelian extension of $k$ contained in $A$. $k$ must be a subfield of the base field of $A$.

`maximal_abelian_subfield`

— Method`maximal_abelian_subfield(K::RelSimpleNumField{AbsSimpleNumFieldElem}; of_closure::Bool = false) -> ClassField`

Using a probabilistic algorithm for the norm group computation, determine the maximal abelian subfield in $K$ over its base field. If `of_closure`

is set to true, then the algorithm is applied to the normal closure of $K$ (without computing it).

## Invariants

`degree`

— Method`degree(A::ClassField)`

The degree of $A$ over its base field, i.e. the size of the defining ideal group.

`base_ring`

— Method`base_ring(A::ClassField)`

The maximal order of the field that $A$ is defined over.

`base_field`

— Method`base_field(A::ClassField)`

The number field that $A$ is defined over.

`discriminant`

— Method`discriminant(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}`

Using the conductor-discriminant formula, compute the (relative) discriminant of $C$. This does not use the defining equations.

`conductor`

— Method`conductor(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}`

Return the conductor of the abelian extension corresponding to $C$.

`defining_modulus`

— Method`defining_modulus(CF::ClassField)`

The modulus, i.e. an ideal of the set of real places, used to create the class field.

`is_cyclic`

— Method`is_cyclic(C::ClassField)`

Tests if the (relative) automorphism group of $C$ is cyclic (by checking the defining ideal group).

`is_conductor`

— Method`is_conductor(C::Hecke.ClassField, m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}=InfPlc[]; check) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}`

Checks if (m, inf_plc) is the conductor of the abelian extension corresponding to $C$. If `check`

is `false`

, it assumes that the given modulus is a multiple of the conductor. This is usually faster than computing the conductor.

`is_normal`

— Method`is_normal(C::ClassField) -> Bool`

For a class field $C$ defined over a normal base field $k$, decide if $C$ is normal over $Q$.

`is_central`

— Method`is_central(C::ClassField) -> Bool`

For a class field $C$ defined over a normal base field $k$, decide if $C$ is central over $Q$.

## Operations

`*`

— Method`*(A::ClassField, B::ClassField) -> ClassField`

The compositum of $a$ and $b$ as a (formal) class field.

`compositum`

— Method`compositum(a::ClassField, b::ClassField) -> ClassField`

The compositum of $a$ and $b$ as a (formal) class field.

`==`

— Method`==(a::ClassField, b::ClassField)`

Tests if $a$ and $b$ are equal.

`intersect`

— Method`intersect(a::ClassField, b::ClassField) -> ClassField`

The intersection of $a$ and $b$ as a class field.

`prime_decomposition_type`

— Method`prime_decomposition_type(C::ClassField, p::AbsNumFieldOrderIdeal) -> (Int, Int, Int)`

For a prime $p$ in the base ring of $r$, determine the splitting type of $p$ in $r$. ie. the tuple $(e, f, g)$ giving the ramification degree, the inertia and the number of primes above $p$.

`is_subfield`

— Method`is_subfield(a::ClassField, b::ClassField) -> Bool`

Determines if $a$ is a subfield of $b$.

`is_local_norm`

— Method`is_local_norm(r::ClassField, a::AbsNumFieldOrderElem) -> Bool`

Tests if $a$ is a local norm at all finite places in the extension implicitly given by $r$.

`is_local_norm`

— Method`is_local_norm(r::ClassField, a::AbsNumFieldOrderElem, p::AbsNumFieldOrderIdeal) -> Bool`

Tests if $a$ is a local norm at $p$ in the extension implicitly given by $r$. Currently the conductor cannot have infinite places.

`normal_closure`

— Method`normal_closure(C::ClassField) -> ClassField`

For a ray class field $C$ extending a normal base field $k$, compute the normal closure over $Q$.

`subfields`

— Method`subfields(C::ClassField; degree::Int, is_normal, type) -> Vector{ClassField}`

Find all subfields of $C$ over the base field.

If the optional keyword argument `degree`

is positive, then only those with prescribed degree will be returned.

If the optional keyword `is_normal`

is given, then only those that are normal over the field fixed by the automorphisms is returned. For normal base fields, this amounts to extensions that are normal over `Q`

.

If the optional keyword `is_normal`

is set to a list of automorphisms, then only those wil be considered.

`type`

can be set to the desired relative Galois group, given as a vector of integers descibing the structure.

This will not find all subfields over $\mathbf{Q}$, but only the ones sharing the same base field.