Two variations of a theorem of Kronecker (Q2573228)

The authors establish two extensions of Kronecker's result, which described algebraic integers in the unit circle. In Theorem 1 they describe algebraic numbers \(\alpha\) of degree \(d\), lying with all their conjugates in the disc \(| z| \leq v^{-1/d}\), where \(v\) is the leading coefficient of the minimal polynomial (over \(\mathbb Z\)) of \(\alpha\). Its proof is based on a result of \textit{R. Robinson} [Math. Z. 110, 41--51 (1969; Zbl 0179.07701)], describing sets of conjugates lying in a circle. The second theorem deals with an arbitrary field \(F\), and shows that if \(\omega\) is an element of the algebraic closure of \(F\), and there exists a polynomial \(P(X_1,\dots,X_d)\) over \(F\), which does not vanish at the origin, and moreover there is a sequence \((n_{j,1},\dots,n_{j,d})\) of \(d\)-tuples of integers with \(\lim_{j\to\infty}d_{j,i}=\infty\) for \(i=1,2,\dots\), such that \(P(\omega^{n_{j,1}},\dots,\omega^{n_{j,d}})=0\), then \(\omega\) is a root of unity. The proof is achieved by a clever reduction to the case of a prime field.