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Function fields form the Magma category FldFun and
function field orders form the Magma category RngFunOrd.
The notional
power structures exist as parents of function fields and their orders
but allow no operations.
Category(F) : FldFun -> Cat
Category(O) : RngFunOrd -> Cat
Parent(F) : FldFun -> Pow
Parent(O) : RngFunOrd -> Pow
More interesting related structures (than above) are listed below.
PrimeField(F) : FldFun -> Rng
PrimeRing(O) : RngFunOrd -> Rng
The prime field of the function field F or the order O (prime ring
of the constant field).
DefiningConstantField(F) : FldFunG -> Rng
The constant field k, where F = k(x, α).
The exact constant field of the algebraic function field F/k, i.e. the
algebraic closure in F of the constant field k of F, together with the
inclusion map.
BaseField(F) : FldFun -> Rng
CoefficientRing(F) : FldFun -> Rng
CoefficientField(F) : FldFun -> Rng
The rational function field k(x) if the function field F is
an extension of k(x) and k if F is an extension of k. If F
is an extension of another algebraic function field then this field
will be returned.
Applies to more general fields within Magma than function fields.
Returns whether G is amongst the recursively defined base fields
of F.
CoefficientRing(O) : RngFunOrd -> Rng
The polynomial algebra k[x] if the order O is finite or the degree valuation ring
if O is infinite. If O is an extension
of another order of an algebraic function field this order will be returned.
BaseField(FF) : FldFunOrd -> Rng
CoefficientRing(FF) : FldFunOrd -> Rng
CoefficientField(FF) : FldFunOrd -> Rng
Given a field of fractions FF of an order O return the field of fractions
of the coefficient ring of O.
For a non equation order O returns the order which O was created as a
transformation of. This order is one transformation closer to the equation
order.
The function field which O is an order of.
FieldOfFractions(FF) : FldFunOrd -> FldFunOrd
FieldOfFractions(F) : FldFun -> FldFun
Given an order O, this function returns the field of fractions, a field
with the same basis as O.
On a function field or a field of fractions this function is trivial.
Given a field of fractions FF return the order O which is the ring of
integers of FF.
The function field F represented as an extension of a rational function
field.
This function gives the representation of function fields F/k as
finite extensions.
The order O as an extension of its bottom coefficient ring, (i.e. the order
of the RationalExtensionRepresentation of the
field of fractions of O corresponding to O).
The function field F expressed as an extension of its constant field.
UnderlyingField(F) : FldFunG -> FldFunG
UnderlyingRing(F, R) : FldFunG, Rng -> FldFunG
UnderlyingField(F, R) : FldFunG, Rng -> FldFunG
Return the underlying ring of the function field F over R. This is F expressed as an
extension of R. If R is not given then it is taken to be the coefficient
field of the coefficient field of F. The field R must appear in the tower
of coefficient fields under F.
Embed(F, L, s) : FldFun, FldFun, [FldFunElt] ->
Install the embedding of F into L with the image(s) of the primitive
element(s) of F being the element a in L or the images in s in L.
Print: RngIntElt Default: 0
Return whether there is an embedding of O1 into O2 already installed.
For information about the installed embedding set the Print parameter
> 0 with higher values providing more information than lower.
Print: RngIntElt Default: 0
Return whether an embedding of O1 into O2 can be computed.
For information about the installed embedding set the Print parameter
> 0 with higher values providing more information than lower.
The set of places of the algebraic function field F/k.
The group of divisors of the algebraic function field F/k.
The space of differentials of the algebraic function field F/k.
> R<x> := FunctionField(GF(5));
> P<y> := PolynomialRing(R);
> f := y^3 + (4*x^3 + 4*x^2 + 2*x + 2)*y^2 + (3*x + 3)*y + 2;
> F<alpha> := FunctionField(f);
> ConstantField(F);
Finite field of size 5
> CoefficientField(F);
Univariate rational function field over GF(5)
Variables: x
> CoefficientRing(MaximalOrderFinite(F));
Univariate Polynomial Ring in x over GF(5)
> FieldOfFractions(IntegralClosure(ValuationRing(R), F));
Algebraic function field defined over Univariate rational function field over
GF(5)
Variables: x by
y^3 + (4*x^3 + 4*x^2 + 2*x + 2)*y^2 + (3*x + 3)*y + 2
> Order(IntegralClosure(ValuationRing(R), F),
> MatrixAlgebra(CoefficientRing(MaximalOrderInfinite(F)), 3)!4,
> CoefficientRing(MaximalOrderInfinite(F))!1);
Maximal Order of F over Valuation ring of Univariate rational function field
over GF(5) with generator 1/x
> SubOrder($1);
Maximal Order of F over Valuation ring of Univariate rational function field
over GF(5) with generator 1/x
> Places(F);
Set of places of F
> DivisorGroup(F);
Divisor group of F
Output from UnderlyingRing is shown.
> PF<x> := PolynomialRing(GF(31, 3));
> P<y> := PolynomialRing(PF);
> FF1<b> := ext<FieldOfFractions(PF) | y^2 - x^3 + 1>;
> P<y> := PolynomialRing(FF1);
> FF2<d> := ext<FF1 | y^3 - b*x*y - 1>;
> RationalExtensionRepresentation(FF2);
Algebraic function field defined over Univariate rational function field over
GF(31^3) by
y^6 + 29*y^3 + (30*x^5 + x^2)*y^2 + 1
> UnderlyingRing(FF2);
Algebraic function field defined over Univariate rational function field over
GF(31^3) by
y^6 + 29*y^3 + (30*x^5 + x^2)*y^2 + 1
> UnderlyingRing(FF2, FieldOfFractions(PF));
Algebraic function field defined over GF(31^3) by
$.1^6 + 29*$.1^3 + 30*$.1^2*$.2^5 + $.1^2*$.2^2 + 1
Reduction: BoolElt Default: true
SetVerbose("WeilRes", n): Maximum: 1
A hyperelliptic function field in the Weil restriction over
GF(q) of the elliptic function field E: y2 + xy + x3 + ax2 +
b defined over GF(qn) where q is a power of 2. Also returns a
function which can be used to map a place (not a pole or zero of x)
of F into a divisor of the result. See [Gau00]. Reduction
indicates whether a (possibly quite expensive) reduction step is
performed at the end of the computation. It defaults to true.
Return the function field with constant field E which contains the function
field F.
The ring E must cover the constant field of F. If E is contained
in the exact constant field of F then F and the new field will be
isomorphic.
Changing the constant field to the exact constant field is shown below.
> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<c> := FunctionField(y^6 + y + 2);
> E<a> := ExactConstantField(F);
> C, r := ConstantFieldExtension(F, E);
> r(c);
1/16*(a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)
> $1 @@ r;
c
> e := Random(C, 2);
> e @@ r;
1/2*x*c^5 - 3*x*c^4 + (-12*x + 8)*c^3 + (-16*x + 24)*c^2 + (-8*x + 16)*c - 1
> r($1);
1/2*(-a^5 - a^2 + a)*$.1 + a^5 + a^4 - a^3 - a^2 - 1
Given an algebraic function field F return a function field which is
isomorphic to F and defined by a monic polynomial.
Given an order O belonging to a function field F, this function returns
the order obtained by applying size-reduction to the basis of O.
Given an order O of an algebraic function field and a prime ideal p
of O, return the localization of O at p and the map from O into the
localization.
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