1,728 research outputs found
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Computation of Galois groups of rational polynomials
Computational Galois theory, in particular the problem of computing the
Galois group of a given polynomial is a very old problem. Currently, the best
algorithmic solution is Stauduhar's method. Computationally, one of the key
challenges in the application of Stauduhar's method is to find, for a given
pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial
F that is H-invariant, but not G-invariant. While generic, theoretical methods
are known to find such F, in general they yield impractical answers. We give a
general method for computing invariants of large degree which improves on
previous known methods, as well as various special invariants that are derived
from the structure of the groups. We then apply our new invariants to the task
of computing the Galois groups of polynomials over the rational numbers,
resulting in the first practical degree independent algorithm.Comment: Improved version and new titl
Maps between curves and arithmetic obstructions
Let X and Y be curves over a finite field. In this article we explore methods
to determine whether there is a rational map from Y to X by considering
L-functions of certain covers of X and Y and propose a specific family of
covers to address the special case of determining when X and Y are isomorphic.
We also discuss an application to factoring polynomials over finite fields.Comment: 8 page
Galois invariant smoothness basis
This text answers a question raised by Joux and the second author about the
computation of discrete logarithms in the multiplicative group of finite
fields. Given a finite residue field \bK, one looks for a smoothness basis
for \bK^* that is left invariant by automorphisms of \bK. For a broad class
of finite fields, we manage to construct models that allow such a smoothness
basis. This work aims at accelerating discrete logarithm computations in such
fields. We treat the cases of codimension one (the linear sieve) and
codimension two (the function field sieve)
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