## 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 $$\displaystyle\max \{\beta_i\} > 2 + 4^{-2n-3}$$.

### MSC:

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

### Keywords:

algebraic integers; house; maximum modulus of conjugates

Zbl 02750398
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