The set of places of the algebraic function field F/k.
A sequence containing all places of F/k lying above the place P of the coefficient field k(x) of F. F must be a finite extension of k(x).
Sequence of tuples of ramification indices and residue degrees of the places of F/k lying over the place P of the coefficient field k(x) of F. F must be a finite extension of k(x).
A sequence containing all zeros of the algebraic function a.
A sequence containing all poles of the algebraic function a.
The place corresponding to the prime ideal I, where I is defined over the `finite' or `infinite' maximal order.
Sequences containing the places and exponents occurring in the divisor D.
Change the print name employed when displaying P to be the first element in the sequence of strings s which must have length 1.
F/k denotes a global function field in this section.
Returns true and a place of degree m if and only if there exists such in F/k; false otherwise.
Returns true and a random place of degree m of F/k (false if there are none).
A sequence containing the places of degree m of F/k.
The sets of function field places form the Magma category PlcFun. The notional power structure exists as parent but allows no operations.
The corresponding function field.
The group of divisors of the algebraic function field F/k, which is the free abelian group generated by the elements of the set of places of F/k.
SeparatingElement: FldFunGElt Default:
The Weierstrass places of F/k. The semantics of calling WeierstrassPlaces() with F/k or the zero divisor of F/k are identical. See the description of tttRef{DivFunElt:WeierstrassPlaces}.
F/k denotes a global function field in this section.
The number of places of degree one in the constant field extension of degree m of F/k. Contrary to the Degree() function the degree is here taken over the respective exact constant fields.
The minimum of the Serre and Ihara bound on the number of places of degree one in the constant field extension of degree m of F/k. Contrary to the Degree() function the degree is here taken over the respective exact constant fields.
The number of places of degree m of F/k. Contrary to the Degree() function the degree is here taken over the respective exact constant fields.
A sequence containing the places of degree m of F/k.
Returns divisors D_1, D_2 such that P = kD_1 + D_2 and the exponents in D_2 are of absolute value less than |k|. The operations div and mod yield D_1 resp. D_2.
Returns true if the place P is a `finite' place.
Whether the degree one place P is a Weierstraß place of its function field F. See the description of tttRef{DivFunElt:WeierstrassPlaces}.
The function field that corresponds to the place P.
The degree of the place P over the constant field of definition k.
The ramification index of the place P over its subplace of the rational function field k(x) (the function field of P must be a finite extension of k(x)).
The degree of inertia (or residue class degree) of a place P over the corresponding subplace of the rational function field (the function field of P must be a finite extension of k(x))
A monic prime polynomial in k[x] or 1/x, corresponding to the place of the rational function field P lies above (the function field of P must be a finite extension of k(x)).
The residue class field of the place P.
Evaluate the algebraic function a at the place P. If it is not defined at P, infinity is returned.
Lift the element a of the residue class field of the place P (including infinity) to an algebraic function.
Two algebraic functions having P as their unique common zero.
A local uniformizing parameter at the place P.
The valuation of the element a at the place P.
Create a prime ideal corresponding to the place P.
> R<x> := FunctionField(GF(9)); > P<y> := PolynomialRing(R); > f := y^4 + (2*x^5 + x^4 + 2*x^3 + x^2)*y^2 + x^8 > + 2*x^6 + x^5 +x^4 + x^3 + x^2; > F<a> := FunctionField(f); > Genus(F); 7 > NumberOfPlaces(F, 2); 28 > _, P := RandomPlace(F, 2); > P; Place of F defined by x^2 + w1^6*x + w1^2 and a + w1*x + 1 > LocalUniformizer(P); x^2 + w1^6*x + w1^2 > TwoGenerators(P); x^2 + w1^6*x + w1^2 a + w1*x + 1 > ResidueClassField(P); Finite field of size 3^4 > Evaluate(1/LocalUniformizer(P), P); Infinity > Valuation(1/LocalUniformizer(P), P); -1