{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 $(\alpha,\alpha)$.


Thue-Siegel rational approximation bound

{c}

For an algebraic number $\alpha$ of degree $d$ and a small $\kappa>0$, <Siegel lemma> constructs integer polynomials $P,Q$ of degree at most $n$ and exponentially bounded coefficients for which $P(X)+\alpha Q(X)$ has a zero at $X=\alpha$ of order close to $2n/d$. A <Wronskian> then prevents every specialization $P(X)+yQ(X)$ from having a high-order zero at a rational point. This is the auxiliary-polynomial mechanism in the Thue-Siegel method.


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 Incoming links: Thue-Siegel auxiliary polynomial