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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.
Approximating Rings of Integers in Number Fields
, 1994
"... In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be wellsolved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large ..."
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Cited by 16 (0 self)
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In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be wellsolved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tame" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a ...
The Totally Real A_5 Extension of Degree 6 with Minimum Discriminant
 Math
, 1998
"... CONTENTS 1. Introduction 2. Generation of Polynomials 3. Processing of Generated Polynomials 4. Summary of Results References Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide `a la Recherche (Quebec). We determ ..."
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Cited by 1 (0 self)
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CONTENTS 1. Introduction 2. Generation of Polynomials 3. Processing of Generated Polynomials 4. Summary of Results References Research supported by the Natural Sciences and Engineering Research Council (Canada) and Fonds pour la Formation de Chercheurs et l'Aide `a la Recherche (Quebec). We determine the totally real algebraic number field F of degree 6 with Galois group A 5 and minimum discriminant, showing that it is unique up to isomorphism and that it is generated by a root of the polynomial f(t) = t 6 \Gamma 10t 4 + 7t 3 + 15t 2 \Gamma 14t + 3 over the rationals. We also list the fundamental units and class number of F , as well as data for several other fields that arose in our computations and that might be of interest. 1. INTRODUCTION The comput
ON INTEGRAL BASES IN RELATIVE QUADRATIC EXTENSIONS
"... Abstract. Let F be an algebraic number field and E a quadratic extension with E = F ( √ µ). We describe a minimal set of elements for generating the integral elements oE of E as an oF module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yiel ..."
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Abstract. Let F be an algebraic number field and E a quadratic extension with E = F ( √ µ). We describe a minimal set of elements for generating the integral elements oE of E as an oF module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of E as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times. 1. Preliminaries The computation of integral bases of algebraic number fields is one of the basic tasks in computational algebraic number theory. Nonetheless, the existing algorithms for this problem tend to be very slow for fields of higher degree. In this paper, we therefore outline a new algorithm for the computation of an integral basis for those fields E which contain a subfield F of index 2. Quadratic extensions have been extensively studied in the past [8, 6] and the main result of this paper is a generalization of a result of Sommer [10], who investigated biquadratic extensions. In the sequel we consider number fields F with [F: Q]=nand E subject to E = F ( √ µ) with an integral nonsquare element µ of F. It is well known that the ring of integers oE of E is not a free oF module, in general. The following theorem gives a necessary and sufficient criterion for the existence of a relative integral basis [1]. Theorem 1.1. (i) Let f(t) =t 2 −µ∈F[t]be the minimal polynomial of √ µ with polynomial discriminant d(f), andletd E/F be the relative discriminant of E/F. A relative integral basis of E/F exists if and only if the ideal Φ satisfying Φ 2 = d(f)d −1 E/F is principal. We call the ideal Φ the index of oF [ √ µ] in oE. (ii) There are always elements ξ1,ξ2,ξ3 ∈oE such that oE = ξ1oF + ξ2oF + ξ3oF.
Informatik der Universität Leipzig
"... On arithmetic and the discrete logarithm problem in class groups of curves Der Fakultät für Mathematik und ..."
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On arithmetic and the discrete logarithm problem in class groups of curves Der Fakultät für Mathematik und