Ideal Operations

pIntegralBasis(I, p): OMIdl, RngElt -> SeqEnum

Given an ideal \(I\) in OM representation and a prime element \(p\) in the field containing \(I\), compute a \(p\)-integral basis for \(I\).

SIntegralBasis(I, S): OMIdl, SeqEnum -> SeqEnum

SIntegralBasis(I, S): RngOrdFracIdl, [RngIntElt] -> SeqEnum

SIntegralBasis(I, S): RngFunOrdIdl, [RngUPolElt] -> SeqEnum

Given an ideal \(I\) in either representation and a sequence \(S\) of primes in the field containing \(I\), compute an \(S\)-integral basis of \(I\) for the given set of primes \(S\).

Basis(I): OMIdl -> SeqEnum

HNF      : BoolElt                    Default: false
Separated: BoolElt                    Default: false

Given an ideal \(I\) in OM representation, return a basis for \(I\).

If HNF is set to true a triangular basis in Hermite form is returned.

If Separated is set to true the basis will be returned as a sequence of numerators and denominators.

Example: Om Ideal Op (ex-a203ed)

> Ax<x> := PolynomialRing(Integers());
> f := x^4 + 12*x^3 + 54*x^2 + 108*x + 89;
> L := NumberField(f);
> p := 2;
> I := OMRepresentation(L,[L.1,p^12]);
> pIntegralBasis(I,p);
[
    1,
    L.1 + 1,
    1/2*(L.1^2 + 2*L.1 + 1),
    1/4*(L.1^3 + 3*L.1^2 + 3*L.1 + 1)
]
> pIntegralBasis(I,p:HNF:=true); // In HNF
[
    1,
    L.1,
    1/2*(L.1^2 + 1),
    1/4*(L.1^3 + L.1^2 + 3*L.1 + 3)
]
> Basis(I);
[
    1,
    L.1 + 1,
    1/2*(L.1^2 + 2*L.1 + 1),
    1/4*(L.1^3 + 3*L.1^2 + 3*L.1 + 1)
]
> Basis(I : HNF := true);
[
    1,
    L.1,
    1/2*(L.1^2 + 1),
    1/4*(L.1^3 + L.1^2 + 3*L.1 + 3)
]

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TwoElement(I): OMIdl -> FldArithElt, FldArithElt

Given an ideal \(I\) in OM representation, return \(a, b\) such that \(e = a*e_1 + b*e_2\) for some \(e_1, e_2\) for all \(e \in I\).

Norm(I): OMIdl -> RngElt

Given an ideal \(I\) in OM representation, compute the norm of \(I\).

Valuation(alpha, P : parameters): FldArithElt, OMIdl -> RngIntElt, FldElt

Valuation(alpha, P : parameters): FldRatElt, OMIdl -> RngIntElt, FldElt

Valuation(alpha, P : parameters): RngIntElt, OMIdl -> RngIntElt, FldElt

Valuation(alpha, P : parameters): RngUPolElt, OMIdl -> RngIntElt, FldElt

RED    : BoolElt                    Default: false,
MoreSFL: BoolElt                    Default: false

Compute the \(P\)-valuation \(v\) of \(\alpha\) at the prime ideal \(P\).

Setting the parameter MoreSFL to true selects a single factor lifting algorithm. Setting the parameter RED to true returns also the class of \(\alpha\) in \(P^v/P^{(v+1)}\).

Valuation(I, P): OMIdl, OMIdl -> RngIntElt

Given ideals \(I\) and \(P\) in OM representation, return the valuation of \(I\) at \(P\).

a mod P: FldArithElt, OMIdl -> FldArithElt

Reduction(a, P): FldArithElt, OMIdl -> FldArithElt

Reduction(a, P, m): FldArithElt, OMIdl, RngIntElt -> [FldArithElt]

Given an element \(a\) of the field containing the prime ideal \(P\), which is in OM representation, return \(a'\) such that \(a = a' + I\) and \(a' \in P^0/P\).

If \(m > 0\) is given then a sequence of length \(m\) of elements in \(P^0/P\) is returned representing the local expansion of \(a\) at \(P\) up to precision \(m\).

Factorization(I): OMIdl -> SeqEnum

Factorisation(I): OMIdl -> SeqEnum

Given an ideal \(I\) in OM representation returns a sequence of tuples of primes \(P_i\) and exponents \(e_i\) such that \(I = \prod_i P_i^{e_i}\).

Example: Om Ideal Ops (ex-b5c67b)

> Ax<x> := PolynomialRing(Integers());
> f := x^5 + 343*x^4 + 49*x^3 + 343*x^2 + 7*x + 6;
> L := NumberField(f);
> I := OMRepresentation(L,[1/L.1^2,12]);
> 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 [
1/36*(7*$.1^4 + 2395*$.1^3 - 1715*$.1^2 + 2107*$.1 - 2009),
12
]

> TwoElement(I);
1
1/36*(91*L.1^4 + 211*L.1^3 + 169*L.1^2 + 175*L.1 + 55)
> Norm(I);
1/36
> Factorization(I);
[
    <OM prime ideal over   2
    of Number Field with defining polynomial x^5 + 343*x^4 + 49*x^3 + 343*x^2 +
    7*x + 6 over the Rational Field
    having residual degree   1
    and ramification index   1
    Last phi polynomial is   x, -2>,
    <OM prime ideal over   3
    of Number Field with defining polynomial x^5 + 343*x^4 + 49*x^3 + 343*x^2 +
    7*x + 6 over the Rational Field
    having residual degree   1
    and ramification index   1
    Last phi polynomial is   x, -2>
]
> Valuation(I, L`PrimeIdeals[2][1]);
-2
> Valuation(I, L`PrimeIdeals[3][1]);
-2

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ResidueField(I): OMIdl -> Fld

Given an ideal \(I\) in OM representation returns the field \(P^0/P\).

Example: Om Ideals Deg Res (ex-7694e8)

> 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);
> L`PrimeIdeals[p];
[
    OM prime ideal over   7
    of Number Field with defining polynomial x^5 + 343*x^4 + 49*x^3 + 343*x^2 +
    7*x + 6 over the Rational Field
    having residual degree   1
    and ramification index   1
    Last phi polynomial is x + 6,
    OM prime ideal over   7
    of Number Field with defining polynomial x^5 + 343*x^4 + 49*x^3 + 343*x^2 +
    7*x + 6 over the Rational Field
    having residual degree   4
    and ramification index   1
    Last phi polynomial is x^4 + x^3 + x^2 + x + 1
]
> ResidueField($1[2]);
Finite field of size 7^4

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