Ideal Arithmetic and Infrastructure in Purely Cubic Function Fields Ideal Arithmetic and Infrastructure in Purely Cubic Function Fields
نویسنده
چکیده
This paper investigates the arithmetic of fractional ideals and the infrastructure of the principal ideal class of a purely cubic function eld of unit rank one. We rst describe how irreducible polynomials split into prime ideals in purely cubic function elds of nonzero unit rank. This decomposition behavior is used to compute so-called canonical bases of fractional ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one, ideal reduction. The paper concludes with an analysis of the infrastructure in the set of reduced fractional principal ideals of a purely cubic function eld of unit rank one.
منابع مشابه
Ideal Arithmetic and Infrastructure in Purely Cubic Function Fields
This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We st...
متن کاملInfrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5
One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In addition, we will give a divisor-theoretic treatment of the infrastructures ofK, including a descript...
متن کاملThe arithmetic of certain cubic function fields
In this paper, we discuss the properties of curves of the form y3 = f(x) over a given field K of characteristic different from 3. If f(x) satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the...
متن کاملComputations in cubic function fields of characteristic three
This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of these function fields, we are able to calculate the field discriminant and the genus. We also give explicit algorithms for ideal arithmetic which for certain cu...
متن کاملReduction in Purely Cubic Function Fields of Unit Rank One
This paper analyzes reduction of fractional ideals in a purely cubic function field of unit rank one. The algorithm is used for generating all the reduced principal fractional ideals in the field, thereby finding the fundamental unit or the regulator, as well as computing a reduced fractional ideal equivalent to a given nonreduced one. It is known how many reduction steps are required to achiev...
متن کامل