### 2-extensions of ℚ with trivial 2-primary Hilbert kernel

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The aim of this note is to offer a summary of the definitions and properties of arithmetic symbols on the linear group Gl(n, F) -F being an arbitrary discrete valuation field- and to show that the natural generalizations of the Parshin symbol on an algebraic surface S to the linear group Gl(n, ΣS) do not allow us to define new 2-dimensional symbols on S.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

We construct an uncountable set of strong automorphisms of the Witt ring of a global field.

Let $p$ be an odd prime and $E/F$ a cyclic $p$-extension of number fields. We give a lower bound for the order of the kernel and cokernel of the natural extension map between the even étale $K$-groups of the ring of $S$-integers of $E/F$, where $S$ is a finite set of primes containing those which are $p$-adic.

Let $p$ be a prime number and $F$ be a number field. Since Iwasawa’s works, the behaviour of the $p$-part of the ideal class group in the ${\mathbb{Z}}_{p}$-extensions of $F$ has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian $p$-group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the ${K}_{0}$ of its ring of integers. The even $K$-groups of rings of integers appear as higher versions of the class group. Many authors...