= Thue-Siegel rational approximation bound
{c}
If a real algebraic number $\alpha$ has degree $d\geq3$, then for every $\varepsilon>0$ there are only finitely many reduced fractions $p/q$ such that
$$
\left|\alpha-\frac pq\right|<q^{-d/2-1-\varepsilon}.
$$
The <Thue-Siegel auxiliary polynomial> proof chooses two such approximations, constructs an auxiliary polynomial nonzero at their pair, and compares its denominator lower bound with the upper bound supplied by its high-order zero at $(\alpha,\alpha)$.
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