We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq [θ] and provide complete algebraic independence results for them.

We discuss the monoidal structure on Franke’s algebraic model for the K(p) -local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers. MSC: 55P42; 55P60; 55U35

For a real number x, we let ⌊x⌉ be the closest integer to x. In this paper, we look at the arithmetic properties of the integers ⌊θ⌉ when n ≥ 0, where θ > 1 is a fixed algebraic number.

We study real algebraic integers larger than 1, whose norm is ±2 and all of whose conjugates have modulus larger than 1. These numbers are characterized in some easy cases.

Real algebraic integers larger than 1 whose minimal polynomials are certain quadrinomials of degree at least 5 with constant term ±2 and all roots outside the closed unit disk are determined and some of their properties are mentioned.

Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F . One of the most fascinating and apparently difficult problems in algebraic K-theory is to compute the groups KiOF . These groups were shown to be finitely generated by Quillen [36] and their ranks were calculated by Borel [7]. Lichtenbaum a...

We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic. §1. Two Conjectures on Diophantine Approximation of Logarithms of Algebraic Numbers In 1953 K. Mahler [7] proved that for any sufficiently large positive integers a and b, the estimates log a ≥ a −40 l...

We construct, for the Weil representation associated with any discriminant form, an explicit basis in which action of involves algebraic integers over its field definition. The a general element $${\text {SL}}_{2}(\mathbb {Z})$$ on many parts these bases is simple and explicit, fact that we use determining dimension space invariants some families forms.

This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integ...

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented...