On a conjecture of Schinzel and Zassenhaus (Q2914888)

An algebraic integer of degree \(n\) is a complex root \(\alpha\) of an irreducible monic polynomial \(P(x)\) of degree \(n\) with integer coefficients. The other roots of \(P(x)\) are called the conjugates of \(\alpha\). The author gives some partial solutions to a conjecture of Schinzel and Zassenhaus. The conjecture asserts that if \(\alpha\not=0\) is an algebraic integer of degree \(n\) and is not a root of unity, then there exists a constant \(c>0\) such that \(|\bar \alpha|\geq 1+\frac cn\), where \(|\bar \alpha| = \max_{1\leq i\leq n} |\alpha_i|\), \(\alpha_1=\alpha\) and \(\alpha_2, \dots, \alpha_n\) are the conjectures of \(\alpha\).