Gelfond-Schneider Theorem/Lemma 4

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Lemma

Let:

$\Delta = \det \left[{\alpha_{i, j}}\right]_{L \times L}$

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 $i, j \in \left\{{1, 2, \ldots, L}\right\}$.

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.


$\Box$