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A survey of computational class field theory

Henri Cohen (1999)

Journal de théorie des nombres de Bordeaux

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

An Invitation to Additive Prime Number Theory

Kumchev, A., Tolev, D. (2005)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

The Littlewood conjecture in Diophantine approximation claims that inf q 1 q · q α · q β = 0 holds for all real numbers α and β , where · denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q 1 q · q α · | q | p = 0 holds for every real number α and every prime number p , where | · | p denotes the p -adic absolute value normalized by | p | p = p - 1 . We survey the known results on these conjectures and highlight recent developments.

Berichtigung.

Paul Bachmann (1905)

Jahresbericht der Deutschen Mathematiker-Vereinigung

Counting discriminants of number fields

Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)

Journal de Théorie des Nombres de Bordeaux

For each transitive permutation group G on n letters with n 4 , we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over such that the Galois group of the Galois closure of L is isomorphic to G .

Diophantine equations after Fermat’s last theorem

Samir Siksek (2009)

Journal de Théorie des Nombres de Bordeaux

These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions:(i)Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem?(ii)Is it useful to combine this approach with traditional approaches to Diophantine equations: Diophantine approximation, arithmetic geometry, ...?

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