# Gelfond-Schneider Theorem/Lemma 4

## 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.

$\Box$