Several types of algebraic numbers on the unit circle

摘要

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.