A refinement of two theorems of Kronecker. (English) Zbl 0128.03402

The authors prove the following two theorems, which are refinements of two results of L. Kronecker [J. Reine Angew. Math. 53, 173–175 (1857; Zbl 02750398)].
Theorem 1. If an algebraic integer \(\alpha\ne 0\) is not a root of unity, and if \(2s\) among its conjugates \(\alpha_i\) \((i = 1, 2, \ldots, n)\) are complex, then \(\displaystyle \max_{1\le i\le n} \vert\alpha_i\vert > 1 + 4^{-s-2}\).
Theorem 2. If a totally real algebraic integer \(\beta\) is different from \(2 \cos \rho\pi\) \((\rho\) rational), and \(\{\beta_i\}\) \((i = 1, 2, \ldots, n)\) is the set of its conjugates, then \(\displaystyle\max \{\beta_i\} > 2 + 4^{-2n-3}\).


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure


Zbl 02750398
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