We show that a cubic algebraic integer over a number field $K,$ with zero trace is a difference of two conjugates over $K$ of an algebraic integer. We also prove that if $N$ is a normal cubic extension of the field of rational numbers, then every integer of $N$ with zero trace is a difference of two conjugates of an integer of $N$ if and only if the $3-$adic valuation of the discriminant of $N$ is not $4.$

In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals $[\frac{r}{s},\frac{r}{s}+\delta ]$, where $\frac{r}{s}$ is a fixed rational and $\delta \to 0$. We also study functions $g_{-},g,g^{+}$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a given interval $I$. We show how to estimate from below the proportion of roots of an integer polynomial which is sufficiently...

Let $M\left(\alpha \right)$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $ℂ$. Then its Lehmer constant$L\left(V\right)$ is inf $M{\left(\alpha \right)}^{1/deg\left(\alpha \right)}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L\left(V\right)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L\left(V\right)$, which we call the transfinite Lehmer constant $L_{\infty }\left(V\right)$.Also, we prove the converse to Langevin’s Theorem, which states that $L\left(V\right)\>1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_{\infty }\left(V\right)$.

Let $\alpha$ be an algebraic integer of degree $d$ with conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\cdots ,{\alpha }_d$. In the paper we give a lower bound for the mean value$$M_p\left(\alpha \right)=\sqrt[p]{\frac{1}{d}{\sum }_{i=1}^d|log|{\alpha }_i{\left|\right|}^p}$$when $\alpha$ is not a root of unity and $p\>1$.

We study the quadratic case of a conjecture made by Van der Geer and Schoof about the behaviour of certain functions which are defined over the group of Arakelov divisors of a number field. These functions correspond to the standard function $h^0$ for divisors of algebraic curves and we prove that they reach their maximum value for principal Arakelov divisors and nowhere else. Moreover, we consider a function $\widetilde{k^0}$, which is an analogue of exp $h^0$ defined on the class group, and we show it also assumes its...

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.

We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form $E:y^2=x^3+c$, where $c$ is an integer.

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.

Let $\alpha$ be a totally positive algebraic integer, with the difference between its trace and its degree at most 6. We describe an algorithm for finding all such $\alpha$, and display the resulting list of 1314 values of $\alpha$ which the algorithm produces.