Given an ideal I in OM representation and a prime element p in the field containing I, compute a p-integral basis for I.
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.
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.
> 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)
]
Given an ideal I in OM representation, return a, b such that e = a * e1 + b * e2 for some e1, e2 for all e ∈I.
Given an ideal I in OM representation, compute the norm of I.
RED: BoolElt Default: false,
MoreSFL: BoolElt Default: false
Compute the P-valuation v of α 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 α in Pv/P(v + 1).
Given ideals I and P in OM representation, return the valuation of I at P.
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' ∈P0/P.If m > 0 is given then a sequence of length m of elements in P0/P is returned representing the local expansion of a at P up to precision m.
Given an ideal I in OM representation returns a sequence of tuples of primes Pi and exponents ei such that I = ∏i Piei.
> 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
Given an ideal I in OM representation returns the field P0/P.
> 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