Ideals in OM Representation#

Ideals of type RngFunOrdIdl can have an OM representation computed from them. Similarly ideals constructed from the Montes algorithm in OM representation can be converted into ideals of type RngFunOrdIdl.

Ideal(I): OMIdl -> RngOrdIdl#: Translates ideal \(I\) in OM representation into a Magma representation.

OMRepresentation(I): RngFunOrdIdl -> OMIdl#
OMRepresentation(I): RngOrdFracIdl -> OMIdl#: Computes the OM representation of the ideal \(I\) in a function field.

OMRepresentation(L, S): FldArith, [FldArithElt] -> OMIdl#
OMRepresentation(L, a): FldArith, FldArithElt -> OMIdl#
OMRepresentation(L, a): FldArith, RngElt -> OMIdl#: Given a function field \(L\) and an element \(a\) or sequence \(S\) of elements in \(L\), construct, in OM representation, the ideal generated by these elements.

Example: Om Rep Ideal (ex-c49fbd)#

> F<t> := FunctionField(Rationals());
> P<x> := PolynomialRing(F);
> L<a> := ext<F | x^2 + t>;
> Montes(L, Numerator(t + 1));
> Ideal(L`PrimeIdeals[t+1][1]);
Prime Ideal of Maximal Equation Order of L over Univariate Polynomial Ring in t
over Rational Field
Generators:
t + 1
a - 1
> OMRepresentation(L, [t^2 + 6*t + 5, (t + 5)*a + t + 5]);
OM ideal of the field Algebraic function field defined over Univariate rational
function field over Rational Field by
x^2 + t
generated by [
t^2 + 6*t + 5,
(t + 5)*a + t + 5
]

> Ideal($1);
Ideal of Maximal Equation Order of L over Univariate Polynomial Ring in t over
Rational Field
Generators:
t^2 + 6*t + 5
(-1/32*t^3 - 9/32*t^2 - 15/32*t + 25/32)*a + 1/32*t^3 + 5/32*t^2 + 7/32*t +
    35/32

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Ideal Arithmetic#

I + J: OMIdl, OMIdl -> OMIdl#
I * J: OMIdl, OMIdl -> OMIdl#
I / J: OMIdl, OMIdl -> OMIdl#: The sum, product or quotient of ideals \(I\) and \(J\) in OM representation.

I ^ n: OMIdl, RngIntElt -> OMIdl#: The \(n\)-th power of the ideal \(I\) in OM representation.

Example: Om Ideal Arith (ex-0a493e)#

> k := GF(13);
> A<t> := PolynomialRing(k);
> Ax<x> := PolynomialRing(A);
> f := x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 +
>     (t^3 + 3*t^2 + 3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t;
> L := FunctionField(f);
> I := OMRepresentation(L,L.1+1);
> I;
OM ideal of the field Algebraic function field defined over Univariate rational
function field over GF(13) by
x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 + (t^3 + 3*t^2 +
3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t
generated by [
$.1 + 1
]

> Montes(L, t + 1);
> P := L`PrimeIdeals[t+1, 1];
>      J := P^-2;
> I*J;
OM ideal of the field Algebraic function field defined over Univariate rational
function field over GF(13) by
x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 + (t^3 + 3*t^2 +
3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t
having the factorization [
<t + 1, 1, -2>,
<t^4 + 10*t^3 + 4*t^2 + t + 7, 1, 1>
]

>  I^2;
OM ideal of the field Algebraic function field defined over Univariate rational
function field over GF(13) by
x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 + (t^3 + 3*t^2 +
3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t
having the factorization [
<t^4 + 10*t^3 + 4*t^2 + t + 7, 1, 2>
]

> I/J;
OM ideal of the field Algebraic function field defined over Univariate rational
function field over GF(13) by
x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 + (t^3 + 3*t^2 +
3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t
having the factorization [
<t + 1, 1, 2>,
<t^4 + 10*t^3 + 4*t^2 + t + 7, 1, 1>
]

> I+J;
OM ideal of the field Algebraic function field defined over Univariate rational
function field over GF(13) by
x^5 + (t^2 + 2*t + 1)*x^4 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x^3 + (t^3 + 3*t^2 +
3*t + 1)*x^2 + (t^4 + 4*t^3 + 6*t^2 + 4*t + 1)*x + t
having the factorization  []
generated by [
1,
0
]

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Ideal Predicates#

IsOne(I): OMIdl -> BoolElt#: Given an ideal \(I\) in OM representation returns whether the ideal is generated by the \(1\) element of the field.

IsZero(I): OMIdl -> BoolElt#: Given an ideal \(I\) in OM representation returns whether the ideal contains only the \(0\) element of the field.

I eq J: OMIdl, OMIdl -> BoolElt#: Given two ideals \(I\) and \(J\) in OM representation returns whether these ideals are the same.

a in I: RngElt, OMIdl -> BoolElt#: Given an element \(a\) coercible into the field containing the ideal \(I\) in OM representation return whether \(a\) is contained in the ideal.

I subset J: OMIdl, OMIdl -> BoolElt#: Given two ideals \(I\) and \(J\) in OM representation returns whether \(I\) is contained in \(J\).

IsPrime(I): OMIdl -> BoolElt#: Given an ideal \(I\) in OM representation returns whether only the \(1\) ideal and \(I\) contain \(I\).

IsIntegral(I): OMIdl -> BoolElt#: Given an ideal \(I\) in OM representation returns whether all elements in the ideal \(I\) are integral.