Properties of Class Fields

Let D be an effective divisor and U be a subgroup of the ray class group Cl_D. The main existence theorem of class field theory asserts that there is exactly one function field corresponding to the quotient Cl_D/U whose Galois group is isomorphic to Cl_D/U in a canonical way.

Since the field is uniquely defined this way so are its invariants. Some of the invariants can easily be read off the groups involved. Therefore none of the functions listed here will compute a set of defining equations. They can therefore be used on very large fields.

Conductor(m) : DivFunElt -> DivFunElt

Let m be an effective divisor. This function computes the conductor of Cl_m which is the smallest divisor f such that the projection Cl_m to Cl_f is surjective.

Conductor(m, U) : DivFunElt, GrpAb -> DivFunElt

Let m be an effective divisor and U be a subgroup of the ray class group of m. This function computes the conductor of Cl_m/U which is the smallest divisor f such that the projection π:Cl_m/U to Cl_f/π(U) is an isomorphism.

Conductor(A) : FldFunAb -> DivFunElt

For an abelian extension A of global function fields, compute its conductor, ie. the conductor of the norm group of A.

DiscriminantDivisor(m, U) : DivFunElt, GrpAb -> DivFunElt

Let m be an effective divisor and U a subgroup of ray class group such that Cl_m/U is finite. The discriminant divisor is defined as the norm of the different divisor.

DiscriminantDivisor(A) : FldFunAb -> DivFunElt

For an abelian extension A of a global function field, compute its discriminant divisor, ie. the norm of the different divisor. Note that the discriminant divisor can be computed from the norm group, thus no defining equation is derived.

DegreeOfExactConstantField(m) : DivFunElt -> RngIntElt

Let m be an effective divisor. Since the ray class field modulo m is always an infinite field extension containing the algebraic closure of the constant field, this returns ∞.

DegreeOfExactConstantField(m, U) : DivFunElt, GrpAb -> RngIntElt

Let m be an effective divisor and U be a subgroup of the ray class group, (see RayClassGroup), modulo m. This function computes the degree of the algebraic closure of the constant field in the class field corresponding to Cl_m/U. This can be infinite.

DegreeOfExactConstantField(A) : FldFunAb -> RngIntElt

The degree of the exact constant field of the abelian extension A of a global function field. This is the degree of the algebraic closure of the constant field of the base field of A in A. The degree of this field can be computed from the norm group, thus no defining equation is derived.

Genus(m, U) : DivFunElt, GrpAb -> RngIntElt

Let m be an effective divisor and U be a subgroup of the ray class group, (see RayClassGroup), modulo m. This function computes the genus of the class field corresponding to Cl_m/U.

Genus(A) : FldFunAb -> RngIntElt

The genus of the abelian extension A of a global function field.

DecompositionType(m, U, p) : DivFunElt, GrpAb, PlcFunElt -> [<f,e>]

Let m be an effective divisor and U be a subgroup of the ray class group, (see RayClassGroup), modulo m such that the quotient Cl_m/U is finite. For a place p this function will determine the decomposition type of the place in the extension defined by Cl_m/U, i.e. it will return a sequence of pairs < f, e > containing the inertia degree and ramification index for all places above p.

DecompositionType(A, p) : FldFunAb, PlcFunElt -> [<f,e>]

For an abelian extension A of a global function field k and a place p of k, compute the degree and ramification index of all places P lying above p.

NumberOfPlacesOfDegreeOne(m, U) : DivFunElt, GrpAb -> RngIntElt

Let m be an effective divisor and U be a subgroup of the ray class group, (see RayClassGroup), modulo m such that the quotient Cl_m/U is finite. This function will compute the number of places of degree 1 that the class field corresponding to Cl_m/U has.

NumberOfPlacesOfDegreeOne(A) : FldFunAb -> RngIntElt

For an abelian extension A of global function fields, compute the number of places of A that are of degree one over the constant field of the base field of A.

Degree(A) : FldFunAb -> RngIntElt

For an abelian extension A of global function fields, return the degree of A over its base ring.

BaseField(A) : FldFunAb -> FldFunG

For an abelian extension A of global function fields, return the base field, ie, the global field k over which A was created as an extension.

A eq B : FldFunAb, FldFunAb -> BoolElt

For two abelian extensions of the same base field, decide if they describe the same field, ie if the norm groups pulled back into a common over group, agree.

A subset B : FldFunAb, FldFunAb -> BoolElt

For two abelian extensions of the same base field, test if the first is contained in the second. This is done by comparing the norm groups in a common over group, thus defining equations are not computed.

A meet B : FldFunAb, FldFunAb -> FldFunAb

Compute the intersection of two abelian extensions of the same base field as an abelian extension.

A * B : FldFunAb, FldFunAb -> FldFunAb

Compute the compositum of two abelian extensions of the same base field as an abelian extension. Since both fields are normal, the compositum is well defined and can be computed from the norm groups alone.