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: Converting to and from OM Representation (ex-738d2c)

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>      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: Arithmetic of OM Ideals (ex-58ff3a)

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> 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.