Given positive real numbers, we prove two inequalities involving their potential energy and power sums. We also a general inequality the discriminant. This can be applied to case of totally irreducible polynomials, although in that more specialised setting results are weaker than those obtained by Smyth's method.

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integers R is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal...

(1) D(α) = D(K)(I(α))2, where I(α) = [OK : Z[α]], the index of the ring generated by α in the full ring of integers OK of K. Classically, the index of the field K is the greatest common divisor of all indices I(α) for α ∈ OK with K = Q(α). Dedekind was the first to show that the index may not be 1 by exhibiting certain cubic and quartic fields with this property. By results of Bauer and von Żyl...

A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the “α + n conjecture” and the “nα conjecture”. These say, respectively, that given any algebraic integer α there is a natural number n such ...