Prime Ideal Factorization in Quartic Number Fields

The American Mathematical Monthly, 2005

2006

The prime ideal decomposition of 2 in a pure quartic field with field index 2 is determined explicitly.</p

Journal of Number Theory, 2016

Mathematics Magazine, 1985

Foundations of Computational Mathematics, Vol. 13, No. 5, 729-762, 2013

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 the defining equation $f(x)$. 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 ideals.

2012

We solve the diophantine equations d1 a 2 d2 a 2 = NF=Q (IF) to decide if an ideal in the ring of integers in a quadratic eld F with hF = 2 is principal or non-principal. As a consequence of this, we distinguish prime and irreducible elements.

Seema

Let A be an integral domain, and let L be a field containing A. An element a of L is said to be integral over A if it is a root of a monic polynomial with coefficients in A, i.e., if it satisfies an equation THEOREM 5.1 The elements of L integral over A form a ring. I shall give two proofs of this theorem. The first uses Newton's theory of symmetric polynomials and a result Eisenstein, and the second is Dedekind's surprisingly modern proof, which avoids symmetric polynomials. FIRST PROOF THAT THE INTEGRAL ELEMENTS FORM A RING A polynomial is said to be symmetric if it is unchanged when its variables are permuted, i.e., if For example are all symmetric. These particular polynomials are called the elementary symmetric polynomials. THEOREM 2.2 (Symmetric function theorem) Let A be a ring. Every symmetric polynomial is equal to a polynomial in the symmetric elementary polynomials with coefficients in A, i.e. , PROOF. We define an ordering on the monomials in the by requiring that if either or equality holds and, for some s,

Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09, 2009

We present an efficient algorithm for factoring a multivariate polynomial f ∈ L[x1,. .. , xv] where L is an algebraic function field with k ≥ 0 parameters t1,. .. , t k and r ≥ 0 field extensions. Our algorithm uses Hensel lifting and extends the EEZ algorithm of Wang which was designed for factorization over Q. We also give a multivariate p-adic lifting algorithm which uses sparse interpolation. This enables us to avoid using poor bounds on the size of the integer coefficients in the factorization of f when using Hensel lifting. We have implemented our algorithm in Maple 13. We provide timings demonstrating the efficiency of our algorithm.

2007

In this thesis, we surveyed the most important methods for fa ctorization of polynomials over a global field, focusing on their strengths and showing their most str iking disadvantages. The algorithms we have selected are all modular algorithms. They rely on the He nsel factorization technique, which can be applied to all global fields giving an output in a local field that can be computed to a large enough precision. The crucial phase of the reconstruction of the ir reducible global factors from the local ones, determines the difference between these algorithms. For di ffe ent fields and cases, different techniques have been used such as residue class computations, ideal cal culus, attice techniques. The tendency to combine ideas from different methods has bee n of interest as it improves the running time. This appears for instance in the latest method due to va n Hoeij, concerning the factorization over a number field. The ideas here can be used over a global function field in the...