Semi-CNS polynomials are monic polynomials with integer coefficients which are related to natural generalizations of the classical decimal representation of the rational integers to algebraic integers. We characterize semi-CNS polynomials of arbitrary degrees thereby extending known results on cubic and irreducible semi-CNS polynomials.

As analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζC(n) at positive integers n. By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius p-th power relations.

Motivated by the search for “almost integers”, we describe the algebraic integers known as Pisot numbers, and explain how they can be used to easily find irrational values that can be arbitrarily close to whole numbers. Some properties of the set of Pisot numbers are briefly discussed, as well as some applications of these numbers to other areas of mathematics.

Using Poonen’s version of “weak vertical method” we produce new examples of “large” and “small” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture on topology of rational points does not hold.