Cantor series constructions of sets of normal numbers
If is a finite Galois extension of number fields with Galois group , then the kernel of the capitulation map of ideal class groups is isomorphic to the kernel of the transfer map where and is the Hilbert class field of . H. Suzuki proved that when is abelian, divides . We call a finite abelian group a transfer kernel for if for some group extension . After characterizing transfer kernels in terms of integral representations of , we show that is a transfer kernel for...
Let with where is a prime number such that or , the fundamental unit of , a prime number such that and , the Hilbert -class field of , the Hilbert -class field of and the Galois group of . According to E. Brown and C. J. Parry [7] and [8], , the Sylow -subgroup of the ideal class group of , is isomorphic to , consequently contains three extensions
Soient où et deux nombres premiers différents tels que , le -corps de classes de Hilbert de , le -corps de classes de Hilbert de et le groupe de Galois de . D’après [4], la -partie du groupe de classes de est de type , par suite contient trois extensions ; . Dans ce papier, on s’interesse au problème de capitulation des -classes d’idéaux de dans
Let be a prime number and be a number field. Since Iwasawa’s works, the behaviour of the -part of the ideal class group in the -extensions of has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian -group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the of its ring of integers. The even -groups of rings of integers appear as higher versions of the class group. Many authors...
An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of to guarantee...