Ideals in OM Representation#

Ideals of type RngOrdFracIdl 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 RngOrdFracIdl.

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 number field.

OMRepresentation(L, S): FldArith, [FldArithElt] -> OMIdl#
OMRepresentation(L, a): FldArith, FldArithElt -> OMIdl#
OMRepresentation(L, a): FldArith, RngElt -> OMIdl#: Given a number 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-738d2c)#

>      Ax<x> := PolynomialRing(Integers());
>      f := x^5 + 343*x^4 + 49*x^3 + 343*x^2 + 7*x + 6;
>      L := NumberField(f);
>      p := 7;
>      Montes(L,p);
> Ideal(L`PrimeIdeals[p,1]);
Prime Ideal
Two element generators:
    [7, 0, 0, 0, 0]
    [6, 1, 0, 0, 0]
> OMRepresentation(L, [7, 1 + L.1 + L.1^2 + L.1^3 + L.1^4]);
OM ideal of the field Number Field with defining polynomial x^5 + 343*x^4 +
49*x^3 + 343*x^2 + 7*x + 6 over the Rational Field
generated by [
7,
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
]

> Ideal($1);
Ideal
Two element generators:
    [7, 0, 0, 0, 0]
    [1, 1, 1, 1, 1]

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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-58ff3a)#

> Ax<x> := PolynomialRing(Integers());
> f := x^5 + 343*x^4 + 49*x^3 + 343*x^2 + 7*x + 6;
> L := NumberField(f);
> Montes(L,7);
> P := L`PrimeIdeals[7,1];
> I := OMRepresentation(L,[L.1]);
> I;
OM ideal of the field Number Field with defining polynomial x^5 + 343*x^4 +
49*x^3 + 343*x^2 + 7*x + 6 over the Rational Field
generated by [
$.1
]

> J := P^-2;
> I*J;
> I^2;
OM ideal of the field Number Field with defining polynomial x^5 + 343*x^4 +
49*x^3 + 343*x^2 + 7*x + 6 over the Rational Field
having the factorization [ <2, 1, 2>, <3, 1, 2> ]

> I/J;
OM ideal of the field Number Field with defining polynomial x^5 + 343*x^4 +
49*x^3 + 343*x^2 + 7*x + 6 over the Rational Field
having the factorization [ <2, 1, 1>, <3, 1, 1>, <7, 1, 2> ]

> I+J;
OM ideal of the field Number Field with defining polynomial x^5 + 343*x^4 +
49*x^3 + 343*x^2 + 7*x + 6 over the Rational Field
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.