Multi-modular algorithm for computing the splitting field of a polynomial

Abstract

Let f be a univariate monic integral polynomial of degree n and let (↵1, . . . , ↵n) be an n-tuple of its roots in an algebraic closureQ of Q. Obtaining an algebraic representation of the splitting field Q(↵1, . . . , ↵n) of f is a question of first importance in e↵ective Galois theory. For instance, it allows us to manipulate symbolically the roots of f . In this paper, we propose a new method based on multi-modular strategy. Actually, we provide algorithms for this task which return a triangular set encoding the splitting ideal of f . We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

Efficient Arithmetic in Successive Algebraic Extension Fields Using Symmetries

“…Such a formula was introduced in [8] for reconstruction of the polynomials g i in general and results in [20] restate it in a Galois theoretical point of view. In [5], a recursive algorithm is given for such an interpolation.…”

“…, α n are all roots of f (x). As described in [20], if a prime p such that proj p (f ) is square-free, p is lucky for the maximal ideal M consitsing of all algebraic relation among α 1 , . .…”

“…In other words, this computation can provide a modular variety V p for a number of lucky primes p for the splitting ideal M of f and the action of G on V p can be synchronized for these lucky primes p (see [20]) and is, by definition, faithful. Thus, Assumption 1 and Assumption 3 hold.…”

Homotopy techniques for multiplication modulo triangular sets

“…Cauchy modules [27] are a basic construction in Galois theory and invariant theory [32,27,1,26]. Starting from a monic polynomial F ∈ R[X] of degree d, we define a triangular set F 1 , .…”

Computation schemes for splitting fields of polynomials

“…In [17,18], new algorithms are proposed for computing the splitting field of a monic irreducible polynomial f with coefficient in K = Q (more generally, these methods can be applied in any global fields). These new algorithms are based on the relationship between the representation of the splitting field by a Gröbner basis and the action of the corresponding Galois group on this basis.…”