Gelfond-Schneider Theorem/Lemma 4
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Lemma
Let:
- $\Delta = \det \sqbrk {\alpha_{i, j} }_{L \times L}$
where the $\alpha_{i, j}$ are algebraic numbers.
Suppose that $T$ is a positive (rational) integer for which $T \alpha_{i, j}$ is an algebraic integer for every $i, j \in \set {1, 2, \ldots, L}$.
Also, suppose that $\Delta \ne 0$.
Then there is a conjugate of $\Delta$ with absolute value $\ge T^{−L}$.
Proof
Observe that $T^L \Delta$ is an algebraic integer so that one of its conjugates has absolute value $\ge 1$.
The result follows.
$\blacksquare$
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