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A note on certain semigroups of algebraic numbers

Maciej Radziejewski (2001)

Colloquium Mathematicae

The cross number κ(a) can be defined for any element a of a Krull monoid. The property κ(a) = 1 is important in the study of algebraic numbers with factorizations of distinct lengths. The arithmetic meaning of the weaker property, κ(a) ∈ ℤ, is still unknown, but it does define a semigroup which may be interesting in its own right. This paper studies some arithmetic(divisor theory) and analytic(distribution of elements with a given norm) properties of that semigroup and a related semigroup of ideals....

A note on circular units in ℤ p -extensions

Radan Kučera (2003)

Journal de théorie des nombres de Bordeaux

In this note we consider projective limits of Sinnott and Washington groups of circular units in the cyclotomic ℤ p -extension of an abelian field. A concrete example is given to show that these two limits do not coincide in general.

A note on free pro- p -extensions of algebraic number fields

Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

For an algebraic number field k and a prime p , define the number ρ to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro- p -group of rank d . The Leopoldt conjecture implies 1 ≤ ρ ≤ r 2 + 1 , ( r 2 denotes the number of complex places of k ). Some examples of k and p with ρ = r 2 + 1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ < r 2 + 1 are given.

A Note on heights in certain infinite extensions of Q

Enrico Bombieri, Umberto Zannier (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of Q . In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a p -adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.

Currently displaying 101 – 120 of 3416