28 research outputs found
A new computational approach to ideal theory in number fields
Let K be the number field determined by a monic irreducible polynomial f(x) with integer coefficients. In previous papers we parameterized the prime ideals of K in terms of certain invariants attached to Newton polygons of higher order of f(x). In this paper we show how to carry out the basic operations on fractional ideals of K in terms of these constructive representations
of the prime ideals. From a computational perspective, these results facilitate
the manipulation of fractional ideals of K avoiding two heavy tasks: the construction
of the maximal order of K and the factorization of the discriminant
of f(x). The main computational ingredient is Montes algorithm, which is an extremely fast procedure to construct the prime idealsPostprint (author’s final draft
A new computational approach to ideal theory in number fields
Let be the number field determined by a monic irreducible polynomial with integer coefficients. In previous papers we parameterized the prime ideals of in terms of certain invariants attached to Newton polygons of higher order of the defining equation . In this paper we show how to carry out the basic operations on fractional ideals of in terms of these constructive representations of the prime ideals. From a computational perspective, these results facilitate the manipulation of fractional ideals of avoiding two heavy tasks: the construction of the maximal order of and the factorization of the discriminant of . The main computational ingredient is Montes algorithm, which is an extremely fast procedure to construct the prime ideals
Computation of Integral Bases
Let be a Dedekind domain, the fraction field of , and
a monic irreducible separable polynomial. For a given non-zero prime ideal
of we present in this paper a new method to compute a
-integral basis of the extension of determined by . Our
method is based on the use of simple multipliers that can be constructed with
the data that occurs along the flow of the Montes Algorithm. Our construction
of a -integral basis is significantly faster than the similar
approach from and provides in many cases a priori a triangular basis.Comment: 22 pages, 4 figure
On the equivalence of types
Types over a discrete valued field are computational objects that
parameterize certain families of monic irreducible polynomials in ,
where is the completion of at . Two types are considered to be
equivalent if they encode the same family of prime polynomials. In this paper,
we characterize the equivalence of types in terms of certain data supported by
them