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24
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. 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 ..."
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Cited by 40 (3 self)
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Abstract. 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. 1.
Using Galois Ideals For Computing Relative Resolvents
 J. Symb. Comp
, 1998
"... . In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, i ..."
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Cited by 30 (4 self)
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. In this paper we establish that some ideals which occur in Galois theory are generated by a triangular set of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may be exploited in order to obtain more efficient algorithms. Actually, it enables us to present here a new algebraic method for computing relative resolvents which works with any kind of invariant. 1. Introduction Let k be a perfect field and ¯ k an algebraic closure of k. Let f be a univariate polynomial of k[X] supposed separable with degree n, and\Omega be an ordered set of the n roots of f in ¯ k n . In [25] is introduced the notion of ideal of\Omega\Gamma32615/39 invariant by a subset L of the symmetric group of degree n. It generalizes the notion of ideal of relations and the notion of ideal of symmetric relations. We call them Galois ideals. This paper presents two important results. First, we prove in Theorem 5.5 that a Galois ideal associated w...
Formal Computation of Galois Groups with Relative Resolvents
 LNCS
, 1995
"... . We propound a systematic and formal method to compute the Galois group of a nonnecessarily irreducible polynomial: we proceed by successive inclusions, using mostly computations on scalars (and very few on polynomials). It is based on a formal method of specialization of relative resolvents: ..."
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Cited by 15 (4 self)
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. We propound a systematic and formal method to compute the Galois group of a nonnecessarily irreducible polynomial: we proceed by successive inclusions, using mostly computations on scalars (and very few on polynomials). It is based on a formal method of specialization of relative resolvents: it consists in expressing the generic coefficients of the resolvent using the powers of a primitive element, thanks to a quadratic space structure; this reduces the problem to that of specializing a primitive element, which we are able to do in the case of the descending by successive inclusions. We incidentally supply a way to make separable a resolvent. 1 Introduction Let f 2 k[T ] be a monic polynomial  which need not be irreducible , with degree n 2, where k is a field of characteristic zero. Let x = (x 1 ; : : : ; xn ) the family of the roots of f in a splitting field of f over k. The Galois group Gal k (f) = Gal(k(x) : k)  that is, by definition, the group of kalgebra...
The Galois theory of periodic points of polynomial maps
 Proc. London Math. Soc. 68
, 1994
"... There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, ..."
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Cited by 11 (0 self)
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There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [1922], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers
The computation of Galois groups over function fields
, 1992
"... Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degre ..."
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Cited by 9 (0 self)
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Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.
Lagrange Resolvents
, 1996
"... This paper is devoted to a sharp investigation of the notion of Lagrange's resolvent and his connections with Galois Theory. 1 Introduction Among the natural questions first encountered in Galois theory, the problem of computing the Galois group of a given polynomial (e.g., of the splitting field of ..."
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Cited by 6 (1 self)
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This paper is devoted to a sharp investigation of the notion of Lagrange's resolvent and his connections with Galois Theory. 1 Introduction Among the natural questions first encountered in Galois theory, the problem of computing the Galois group of a given polynomial (e.g., of the splitting field of this polynomial, relatively to the base field) is quite natural. We will call it the direct Galois problem. The early mathematicians concerned with the socalled Galois theory created the concept of resolvent, a very suitable one in Galois direct problem. In fact, resolvents and Galois groups were discovered simultaneously, essentially by Lagrange, whose papers on algebraic equations really contains reasonings about groups, although Lagrange did not clearly define groups. In these papers, the correspondence between groups and resolvents is also displayed, rather more deeply than in Galois himself (the main contribution of Galois seems related to groups themselves, not to resolvents; he certainly knew Lagrange's work, but he did not have enough time to develop all ideas arising from this knowledge ). During the XIXth century, no really new idea about algebraic equations was found after those of Lagrange, Abel and Galois, and in most papers or monographs, Galois theory was developed using resolvents and nothing else. In the wellknown works of Kronecker, Vogt, Weber, C. Jordan and others, attempts very similar one another in order to determine the Galois group of a polynomial are found: one begins with a
Computing the maximal subgroups of a permutation group I
, 2001
"... We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach ..."
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Cited by 5 (2 self)
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We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach
Generalized explicit descent and its application to curves of genus 3
, 2010
"... Abstract. We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologicallydefined Selmer ..."
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Cited by 3 (2 self)
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Abstract. We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologicallydefined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over Q of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus3 examples defined by polynomials with small coefficients. 1.
A modular method to compute the splitting field of a polynomial. Communication privée
 Jussieu, 75252 Paris Cedex 05 Email address: annick.valibouze@upmc.fr wwwspiral.lip6.fr/˜avb
, 1999
"... Abstract. We provide a modular method for computing the splitting field Kf of an integral polynomial f by suitable use of the byproduct of computation of its Galois group Gf by padic Stauduhar’s method. This method uses the knowledge of Gf with its action on the roots of f over padic number field, ..."
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Cited by 3 (1 self)
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Abstract. We provide a modular method for computing the splitting field Kf of an integral polynomial f by suitable use of the byproduct of computation of its Galois group Gf by padic Stauduhar’s method. This method uses the knowledge of Gf with its action on the roots of f over padic number field, and it reduces the computation of Kf to solving systems of linear equations modulo some power of the chosen prime p. We provide a careful treatment on reducing computational difficulty. We examine the ability/practicality of the method by experiments on a real computer and we study its complexity. 1