On the Computation of the Class Number of an Algebraic Number Field

Buchmann, Johannes; Williams, H. C.

doi:10.2307/2008730

Cited by 5 publications

(1 citation statement)

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“…where h = # Cl O and R = reg O * are the true class number and regulator, and h ′ = # coker φ and R ′ = reg ker φ the conjectured ones; here we assume that H contains all roots of unity in K, which can easily be accomplished [56, §5.4]. Now suppose that we are able to estimate hR up to a factor 2, i.e., that we can compute a number a with a/2 < hR < a; if one assumes the generalized Riemann hypothesis this can probably be done by means of a good algorithm, as in [16]. Then we see from ( Of course, one also wants to know how the running time depends on n, and which value can be taken for the O-constant.…”

Section: Class Groups and Unitsmentioning

confidence: 99%

Algorithms in Algebraic Number Theory

Lenstra¹

1992

Bull. Amer. Math. Soc.

114

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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. IntroductionThe main interest of algorithms in algebraic number theory is that they provide number theorists with a means of satisfying their professional curiosity. The praise of numerical experimentation in number theoretic research is as widely sung as purely numerological investigations are indulged in, and for both activities good algorithms are indispensable. What makes an algorithm good unfortunately defies definition-too many extra-mathematical factors affect its practical performance, such as the skill of the person responsible for its execution and the characteristics of the machine that may be used.The present paper addresses itself not to the researcher who is looking for a collection of well-tested computational methods for use on his recently acquired personal computer. Rather, the intended reader is the perhaps imaginary pure mathematician who feels that he makes the most of his talents by staying away from computing equipment. It will be argued that even from this perspective the study of algorithms, when considered as objects of research rather than as tools, offers rich rewards of a theoretical nature.The problems in pure mathematics that arise in connection with algorithms have all the virtues of good problems. They are of such a distinctly fundamental nature that one is often surprised to discover that they have not been considered earlier, which happens even in well-trodden areas of mathematics; and even in areas that are believed to be well-understood it occurs frequently that the existing theory offers no ready solutions, fundamental though the problems may be. Solutions that have been found often need tools that at first sight seem foreign to the statement of the problem.

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Section: Class Groups and Unitsmentioning

confidence: 99%

Algorithms in Algebraic Number Theory

Lenstra¹

1992

Bull. Amer. Math. Soc.

114

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Quantum Algorithms for Number Fields

Haase

Maier

2007

Elements of Quantum Information

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Quantum algorithms for number fields

Haase¹,

Maier²

2006

Fortschritte der Physik

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This is a survey of recent results on quantum algorithms for the computation of invariants of number fields, namely the class number and the regulator. Most known classical algorithms for the computation of these values are of subexponential complexity and depend on the truth of a still unproven hypothesis of analytic number theory. We use an important number theoretic concept, Minkowski's Geometry of Numbers, to visualize these invariants, and describe the quantum algorithms developed by Hallgren, Schmidt and Vollmer which compute these invariants using a polynomial number of steps. Computational techniques in number fields, which are necessary to justify the classical part of these quantum algorithms, are the focus of the research of our project group, and are explained in detail.

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On the Computation of the Class Number of an Algebraic Number Field

Cited by 5 publications

References 10 publications

Algorithms in Algebraic Number Theory

Algorithms in Algebraic Number Theory

Quantum Algorithms for Number Fields

Quantum algorithms for number fields

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