Several types of algebraic numbers on the unit circle

Let A _p ⊂ C denote the set of all algebraic numbers such that α ∈ A _p if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A _p . In particular, the intersection of A _p and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A _p . Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A _p if and only if no real root of p(X) is positive.Received: 19 November 2004; revised: 9 February 2005