= Solution
The <Liouville approximation theorem> states that if a real irrational algebraic number $\alpha$ has degree $d$, there is $c>0$ such that $|\alpha-p/q|\geq c/q^d$ for every integer $p$ and positive integer $q$.
Choose its irreducible <polynomial> $F\in\mathbb Z[X]$ of degree $d$. Since $\alpha$ is irrational, $F(p/q)\ne0$; after multiplication by $q^d$ it is a nonzero integer. Hence $|F(p/q)|\geq q^{-d}$. On $[\alpha-1,\alpha+1]$ the <polynomial> derivative has a finite bound $M>0$. For $|p/q-\alpha|<1$, the <mean value theorem> gives $|F(p/q)|\leq M|p/q-\alpha|$, proving the required bound with $c=1/M$. For more distant rationals, the bound holds with $c\leq1$. Taking $c=\min\{1,1/M\}$ covers every rational and proves the theorem.
Back to article page
