Ideals for function field orders O are O-modules I subseteq F for which there is a d in F such that dI subseteq O is a non-zero ideal of O, that is they are fractional ideals of O. Over the coefficient ring of O they are also free modules of rank n, where n equals the degree [F:k(x, alpha_1, ..., alpha_r)].
Given an order O, as well as elements a_1, a_2, ..., a_m coercible into the field of fractions F of O, create the fractional ideal of O generated by these elements.
Note that, contrary to the general case for the constructors, the right hand side elements are not necessarily contained in the left hand side.
The ideal of the order O of an algebraic function field whose basis is the matrix or dedekind module T over the coefficient ring of O divided by the element d of the denominator ring of O.
The ideal of the order O of an algebraic function field whose basis if the matrix T over the coefficient ring of O along with the coefficient ideals I_1, ..., I_n or those in S.
Create the ideal x * O where x is coercible into the function field F.
Create a prime ideal corresponding to the place P.
Create two ideals of the `finite' and `infinite' maximal order respectively corresponding to the divisor D.
Return the ideal I as an ideal of O.
The principal ideal generated by c divided by I.
The colon ideal [I:J] of elements which multiply all elements of J into I.
Return whether the ideal I has an nth root and if so return an nth root.
Return the nth root of the ideal I.
Return whether the ideal I is a square and if so return a square root.
Return a square root of the ideal I.
> P<x> := PolynomialRing(GF(79));
> P<y> := PolynomialRing(P);
> Fa<a> := FunctionField(y^2 - x);
> P<y> := PolynomialRing(Fa);
> Fb<b> := FunctionField(y^2 - a);
> P<y> := PolynomialRing(Fb);
> Fc<c> := FunctionField(y^2 + a*b);
> I := a*b*c*MaximalOrderInfinite(Fc);
> IsSquare(I^2);
true Fractional ideal of Maximal Order of Fc over Maximal Order of Fb over
Maximal Equation Order of Fa over Valuation ring of Univariate rational function
field over GF(79) with generator 1/x
Basis:
Pseudo-matrix over Maximal Order of Fb over Maximal Equation Order of Fa over
Valuation ring of Univariate rational function field over GF(79) with generator
1/x
Fractional ideal of Maximal Order of Fb over Maximal Equation Order of Fa over
Valuation ring of Univariate rational function field over GF(79) with generator
1/x
Generators:
1
((78*x^2 + 71*x + 78)/x^2*a + (23*x^2 + 15*x + 78)/x^2)*b + (67*x^2 + 60*x +
78)/x^2*a + (3*x^2 + 47*x + 78)/x * ( 1 0 )
Fractional ideal of Maximal Order of Fb over Maximal Equation Order of Fa over
Valuation ring of Univariate rational function field over GF(79) with generator
1/x
Generators:
1
((59*x^2 + 44*x + 78)/x^2*a + (4*x^2 + 48*x + 78)/x^2)*b + (29*x^2 + 46*x +
78)/x^2*a + (71*x + 78)/x * ( 0 1 )
> _, II := 1;
> II eq I;
true
> MaximalOrderFinite(Fc)!!I;
Ideal of Maximal Order of Fc over Maximal Equation Order of Fb over Maximal
Equation Order of Fa over Univariate Polynomial Ring in x over GF(79)
Generators:
a * b * c
a * b * c
1;
> II eq I;
true
> MaximalOrderFinite(Fc)!!I;
Ideal of Maximal Order of Fc over Maximal Equation Order of Fb over Maximal
Equation Order of Fa over Univariate Polynomial Ring in x over GF(79)
Generators:
a*b*c
a*b*c
Returns true if and only if the ideal I is the zero ideal of the order O.
Returns true if and only if the ideal I is the identity ideal of the order O, i.e. I = O.
Returns true if and only if the ideal I is integral (a true ideal of its order).
Returns true if and only if the ideal I is prime (the order of the ideal must be maximal).
Returns true and a generator if the fractional ideal I is principal, false otherwise. The function field has to be global.
Return true if the inertia degree of P is the degree of its order.
Return true if there is an inert ideal in O above P.
Return true if the ramification index of P is not 1.
Return true if there is a ramified ideal in O above P.
Return true if P is not the only ideal lying above the prime ideal it lies above.
Return true if there are at least 2 distinct ideals which lie in O above P.
Return whether P is not wildly ramified.
Return whether P is not wildly ramified in O.
Return whether the ramification index of P is the same as the degree of its order over its coefficient order.
Return whether there are any totally ramified ideals in O lying above P.
Return whether there are as many ideals as the degree of the order of P lying over the prime P lies over.
Return whether there are as many ideals of O which lie above P as the degree of O.
Return whether the ramification index of P is 1.
Return whether all the ideals of O which lie above P are unramified.
Return whether the ramfication index of P is a multiple of the characteristic of the residue field of P.
Return whether any of the ideals of O which lie above P are wildly ramified.
The completion of the algebraic function field F or an order O of such at the prime ideal p of an order of F or O. The place at the ideal p must have degree 1. The map from F or O into the series ring is returned also.
The intersection of the ideals I and J.
Given invertible ideals of an order O, returns the greatest common divisor of the ideals I and J.
Given invertible ideals of an order O, returns the least common multiple of the ideals I and J.
Factorization of the ideal I (as sequence of prime ideal, exponent pairs). The order must be maximal.
A sequence containing all prime ideals of the order O lying above the prime element or ideal p of the coefficient ring of O.
A sequence containing all prime ideals of the `infinite' maximal order O.
Sequence of tuples of residue degrees and ramification indices of the prime ideals of the order O lying over p, a prime polynomial or ideal or element of valuation ring of valuation 1.
Sequence of tuples of residue degrees and ramification indices of the prime ideals of the `infinite' maximal order O.
Returns the multiplicator ring of the ideal I of the order O, that is, the subring of elements of the field of fractions of O that multiply I into itself.
Al: MonStgElt Default: "Auto"
The p-maximal over order of O where p is a prime ideal of the coefficient ring of O. The parameter Al has the same options as for MaximalOrder.
Returns the p-radical of an order O for a prime ideal p of the coefficient ring of O, defined as the ideal consisting of elements of O for which some power lies in the ideal pO.It is possible to call this function even if p is not prime. In this case the p-trace-radical will be computed, i.e. { x in F | Tr(xO)subseteq C} for F the field of fractions of O and C the order of p (if p is an ideal) or the parent of p otherwise. If p is square free and all divisors are larger than the field degree, this is the intersection of the radicals for all l dividing p.
The valuation of a or I at the prime ideal P. The element a must be coercible into the field of fractions of P's order.
The order of the ideal I.
The "smallest" element d of the coefficient ring of the ideal's order O such that dI subseteq O.
A generator m of the ideal R intersect dI where R is the coefficient ring of the ideal's order and d is the denominator of the ideal I (d is is the second return value).
The intersection of the ideal I with a coefficient ring R of its order.
The integral ideal dI and d, where d is the denominator of I.
The norm of the ideal I, as element of the coefficient field of the algebraic function field to which I belongs.
Given an ideal I with function field F as the field of fractions of its order O, returns two elements a, b in F such that I = aO + bO.
Given a (fractional) ideal I of the order O, return a sequence of elements of the function field F that generate I as an ideal.
A basis of the ideal I as a free module over the coefficient ring of it's order.
Let (b_1, ..., b_n) be the basis of I and let (omega_1, ..., omega_n) be the basis of the order O. A matrix B with coefficients in the rational function field is returned such that (b_1, ..., b_n) = (omega_1, ..., omega_n) B^t.
Let (b_1, ..., b_n) be the basis of I and let (omega_1, ..., omega_n) be the basis of the order O. A matrix T with coefficients in the coefficient ring of O and a denominator d are returned such that (b_1, ..., b_n) = (omega_1, ..., omega_n) T^t / d.
The different of the (possibly fractional) ideal I of an order of an algebraic function field.
The divisor corresponding to the ideal factorization of I.
The divisor corresponding to the ideal factorization of the ideals I and J belonging to the `finite' and `infinite' maximal order.
> PR<x> := FunctionField(Rationals());
> P<y> := PolynomialRing(PR);
> FR1<a> := FunctionField(y^3 - x + 1/x^3);
> P<y> := PolynomialRing(FR1);
> FR2<c> := FunctionField(y^2 - a/x^3*y + 1);
> I := ideal<MaximalOrderFinite(FR2) |
> [ x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
> 468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237, x^9 + 1639*x^8 +
> 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 + 468363837*x^3 +
> 242625823*x^2 + 68744019*x + 8052237, (x^15 + 1639*x^14 + 863249*x^13 +
> 148609981*x^12 + 404988066*x^11 + 567876948*x^10 + 468363837*x^9 +
> 242625823*x^8 + 68744019*x^7 + 8052237*x^6)*c, (x^15 + 1639*x^14 +
> 863249*x^13 + 148609981*x^12 + 404988066*x^11 + 567876948*x^10 +
> 468363837*x^9 + 242625823*x^8 + 68744019*x^7 + 8052237*x^6)*c ]>;
> I;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 * ( 1 0 )
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Generator:
(x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237)/x^2 * ( 0 1 )
> J := ideal<MaximalOrderFinite(FR2) |
> [ x^3 + 278*x^2 + 164*x + 742, x^3 + 278*x^2 + 164*x + 742, (x^9 + 278*x^8 +
> 164*x^7 + 742*x^6)*c, (x^9 + 278*x^8 + 164*x^7 + 742*x^6)*c ]>;
> J;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^3 + 278*x^2 + 164*x + 742 * ( 1 0 )
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Generator:
(x^3 + 278*x^2 + 164*x + 742)/x^2 * ( 0 1 )
> Generators(J);
[
x^3 + 278*x^2 + 164*x + 742,
x^3 + 278*x^2 + 164*x + 742,
(x^7 + 278*x^6 + 164*x^5 + 742*x^4)*c,
(x^7 + 278*x^6 + 164*x^5 + 742*x^4)*c
]
> TwoElement(J);
x^3 + 278*x^2 + 164*x + 742
((3/2*x^10 + 419*x^9 + 802*x^8 + 1441*x^7 + 1484*x^6)*a^2 + (3/2*x^8 + 417*x^7 +
246*x^6 + 1113*x^5)*a + (3/2*x^8 + 837/2*x^7 + 663*x^6 + 1359*x^5 +
1113*x^4))*c + (x^6 + 277*x^5 - 114*x^4 + 578*x^3 - 742*x^2)*a^2 + (3*x^5 +
834*x^4 + 492*x^3 + 2226*x^2)*a - x^3 - 278*x^2 - 164*x - 742
> Minimum(I);
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 1
> Basis(J);
[ 1, x^6*c ]
> Basis(I);
[ 1, x^6*c ]
> I eq J;
false
> II, d := IntegralSplit(I);
> II;
Ideal of Maximal Order of FR2 over Maximal Order of FR1 over Univariate
Polynomial Ring in x over Rational Field
Basis:
Pseudo-matrix over Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x over Rational
Field
Generator:
x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237 * ( 1 0 )
Fractional ideal of Maximal Order of FR1 over Univariate Polynomial Ring in x
over Rational Field
Generator:
(x^9 + 1639*x^8 + 863249*x^7 + 148609981*x^6 + 404988066*x^5 + 567876948*x^4 +
468363837*x^3 + 242625823*x^2 + 68744019*x + 8052237)/x^2 * ( 0 1 )
> d;
1
> IsIntegral(I);
true
> GCD(I, J)*LCM(I, J) eq I*J;
true
The ramification index of the prime ideal I over the corresponding prime of its coefficient ring.
The residue class degree (inertia degree) of the prime ideal I over the corresponding prime of its coefficient ring.
The residue class field of the prime ideal I and the residue class mapping.
The place corresponding to the prime ideal I, where I is defined over the `finite' or `infinite' maximal order.
> R<x> := FunctionField(GF(3)); > P<y> := PolynomialRing(R); > f := y^4 + x*y + x^4 + x + 1; > F<a> := FunctionField(f); > O := MaximalOrderFinite(F); > x*O; Ideal of O Generator: x > L := Factorization(x*O); > L; [ <Ideal of O Generators: x a^2 + a + 2, 1>, <Ideal of O Generators: x a^2 + 2*a + 2, 1> ] > P1 := L[1][1]; > P2 := L[2][1]; > BasisMatrix(P1); [x 0 0 0] [0 x 0 0] [2 1 1 0] [1 1 0 1] > P1 meet P2 eq x*O; true > IsPrime(P1); true > Place(P1); (x, a^2 + a + 2)