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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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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.
Discriminants of Hermitian R[G]modules and Brauer’s Class Number Relation
"... Abstract. The purpose of this paper is to lay the foundations for a quantitative theory of relations among discriminants of hermitian RGmodules which are induced by character relations. This is accomplished by introducing an invariant δ(M) attached to an RGmodule M which plays the role of a corre ..."
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Abstract. The purpose of this paper is to lay the foundations for a quantitative theory of relations among discriminants of hermitian RGmodules which are induced by character relations. This is accomplished by introducing an invariant δ(M) attached to an RGmodule M which plays the role of a correction term in such relations and to study its functorial properties such as localization and induction theorems, behaviour with respect to exact sequences, triviality etc. By means of this formalism it is shown that this invariant may be computed in many cases. An application of this invariant is the class number relation of R. Brauer (1951) and, by using the formalism mentioned above, also that of Dirichlet (1842). 1.
Dyson’s Theorem for curves.
, 811
"... ABSTRACT. Let K be a number field and X1 and X2 two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson Theorem for the product X1×X2. If X i = P1 we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on ..."
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ABSTRACT. Let K be a number field and X1 and X2 two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson Theorem for the product X1×X2. If X i = P1 we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and MordellWeil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem. 1 Introduction. After the proof of the Mordell conjecture by Faltings (the first proof is in [Fa1], but [Fa2], [B2] and [Vo2] are nearer to the spirit of this paper), most of the qualitative results in the diophantine approximation of algebraic divisors by rational points over curves are solved.