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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....

Counting invertible matrices and uniform distribution

Christian Roettger (2005)

Journal de Théorie des Nombres de Bordeaux

Consider the group SL 2 ( O K ) over the ring of algebraic integers of a number field K . Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let SL 2 ( O K , t ) be the number of matrices in SL 2 ( O K ) with height bounded by t . We determine the asymptotic behaviour of SL 2 ( O K , t ) as t goes to infinity including an error term, SL 2 ( O K , t ) = C t 2 n + O ( t 2 n - η ) with n being the degree of K . The constant C involves the discriminant of K , an integral depending only on the signature of K , and the value of the Dedekind zeta function...

Counting irreducible polynomials over finite fields

Qichun Wang, Haibin Kan (2010)

Czechoslovak Mathematical Journal

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: π ( x ) = q q - 1 x log q x + q ( q - 1 ) 2 x log q 2 x + O x log q 3 x , x = q n where π ( x ) denotes the number of monic irreducible polynomials in F q [ t ] with norm x .

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