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On 2 -class field towers of imaginary quadratic number fields

Franz Lemmermeyer (1994)

Journal de théorie des nombres de Bordeaux

For a number field k , let k 1 denote its Hilbert 2 -class field, and put k 2 = ( k 1 ) 1 . We will determine all imaginary quadratic number fields k such that G = G a l ( k 2 / k ) is abelian or metacyclic, and we will give G in terms of generators and relations.

On Bilinear Structures on Divisor Class Groups

Gerhard Frey (2009)

Annales mathématiques Blaise Pascal

It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory.In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures.Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups...

On p 2 -Ranks in the Class Field Tower Problem

Christian Maire, Cam McLeman (2014)

Annales mathématiques Blaise Pascal

Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the p 2 -rank of the class group as a quantity of relevance in the p -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups...

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