The characteristic of the function field F/k or one of its orders O.
Applies to any field in Magma. Returns whether F is perfect.
The degree [F:G] of the field extension F/G where G is the base field of F unless specified. For an order O, this function returns the rank of O as a module over its coefficient ring. Note that this rank is equal to the degree [F:G] where F and G are the field of fractions of O and the coefficient ring of O respectively.
The degree of the function field F or the order O as a finite extension of k(x) or k[x] or as an infinite extension of k.
The defining polynomial of the function field F over its coefficient ring. For an order O belonging to a function field F, this function returns the defining polynomial of O, which may be different from that of F/k(x, α1, ..., αr).
Return the defining polynomials of the function field F or the order O as a sequence of polynomials over the coefficient ring.
The basis 1, α, ..., αn - 1 of the function field F[α] over the coefficient field.Given an order O belonging to a function field F, this function returns the basis of O in the form of function field elements.
Given an additional ring R, return the basis of O as elements of R.
Return the matrix M and a denominator d which transforms elements of the order O1 into elements of the order O2.
The coefficient ideals of the order O of a relative extension. These are the ideals {Ai} of the coefficient ring of O such that for every element e of O, e = ∑i ai * bi where {bi} is the basis returned for O and each ai ∈Ai.
Given an order O in a function field F of degree n, this returns an n x n matrix whose i-th row contains the coefficients for the i-th basis element of O with respect to the power basis of F. Thus, if bi is the i-th basis element of O, bi=∑j=1nMijαj - 1 where M is the matrix and α is the generator of F.
A root of the defining polynomial of the order O.
The discriminant of the order O, up to a unit in its coefficient ring.
The discriminant of the order O of an algebraic function field F over the bottom coefficient ring of O, (the subring of the rational function field F extends).
The dimension of the exact constant field of the function field F/k over k. The exact constant field is the algebraic closure of k in F.
Al: MonStgElt Default:
IsExact: BoolElt Default: false
The genus of the function field F/k. If F is an extension of a rational function field over Q or Fq by a single monic integral polynomial and the parameter Al is set to "Montes" then the Montes algorithm [Sta18] will be used to compute the genus.The index [k0:(ConstantField)(F)], where k0 is the full constant field, is also returned. If it is known beforehand that k0 is the constant field F is defined over then set the parameter IsExact := true.
> 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>;
> Characteristic(FF2);
31
> EFF2I := EquationOrderInfinite(FF2);
> MFF2I := MaximalOrderInfinite(FF2);
> Degree(MFF2I) eq 3;
true
> AbsoluteDegree(EFF2I);
6
> Genus(FF2);
9
> DefiningPolynomial(EFF2I);
$.1^3 + [ 0, 30/x^3 ]*$.1 + [ 30/x^9, 0 ]
> Basis(MFF2I);
[ 1, 1/x*d, 1/x^2*d^2 ]
> Discriminant(EFF2I);
Ideal of Maximal Equation Order of FF1 over Valuation ring of Univariate
rational function field over GF(31^3)
Variables: x with generator 1/x
Generator:
(4*x^3 + 27)/x^15*b + 4/x^18
> AbsoluteOrder(EFF2I);
Order of 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 over Valuation ring of Univariate rational
function field over GF(31^3) with generator 1/x
> AbsoluteDiscriminant(EFF2I);
(2*x^9 + 25*x^6 + 6*x^3 + 29)/x^33
> Discriminant($2);
(30*x^24 + 6*x^21 + 16*x^18 + 20*x^15 + 16*x^12 + 7*x^9 + 27*x^6 + 3*x^3 +
30)/x^48
> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<a, b> := FunctionField([3*y^3 - x^2, x*y^2 + 1]);
> DefiningPolynomials(F);
[
3*y^3 - x^2,
x*y^2 + 1
]
> DefiningPolynomials(EquationOrderFinite(F));
[
y^3 - 1/3*x^2,
y^2 + x
]
> DefiningPolynomials(EquationOrderInfinite(F));
[
$.1^3 - 1/3/$.1^4,
$.1^2 + 1/$.1
]
> Basis(F);
[
1,
a,
a^2,
$.1*b,
$.1*a*b,
$.1*a^2*b
]
> TransformationMatrix(EquationOrderFinite(F), MaximalOrderFinite(F));
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 x 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 x 0]
[0 0 0 0 0 x]
1
> TransformationMatrix(MaximalOrderFinite(F), EquationOrderFinite(F));
[x 0 0 0 0 0]
[0 x 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 x 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
x
SeparatingElement: FldFunGElt Default:
The sequence of global gap numbers of the function field F/k (in characteristic zero this is always [1, ..., g]). A separating element used internally for the computation can be specified, it defaults to SeparatingElement(F). See the description of GapNumbers.
The sequence of gap numbers of the function field F/k at P where P must be a place of degree one. See the description of GapNumbers.
Returns a separating element of the function field F/k.
SeparatingElement: FldFunGElt Default:
The ramification divisor of the function field F/k. The semantics of calling RamificationDivisor() with F or the zero divisor of F are identical. For further details see the description of RamificationDivisor.
SeparatingElement: FldFunGElt Default:
The Weierstrass places of the function field F/k. The semantics of calling WeierstrassPlaces with F or the zero divisor of F are identical. See the description of WeierstrassPlaces.
SeparatingElement: FldFunGElt Default:
The Wronskian orders of the function field F/k. The semantics of calling WronskianOrders with F or the zero divisor of F are identical. See the description of WronskianOrders.
The different of the maximal order O.
The index of S in O where S is a suborder of O and O and S have the same equation order.