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A generalization of Dirichlet's unit theorem

Paul Fili, Zachary Miner (2014)

Acta Arithmetica

We generalize Dirichlet's S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a ℚ-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over ℚ retain their linear independence...

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 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 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 Sinnott's index formula

Kazuhiro Dohmae (1997)

Acta Arithmetica

Let k be an (imaginary or real) abelian number field whose conductor has two distinct prime divisors. We shall construct a basis for the group C of circular units in k and compute the index of C in the group E of units in k. This result is a generalization of Theorem 3.3 in a previous paper [1].

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