 Proof of Liouville approximation theorem

Let $\alpha$ be a real irrational algebraic number of field extension degree $d\ge 2$, and let $P\in \mathbb{Z}[X]$ be its primitive algebraic minimal polynomial. For any rational $p/q$ with positive integer $q$, irreducibility gives $P(p/q)\ne 0$. Consequently the nonzero integer $q^{d}P(p/q)$ givesOn a fixed closed interval around $\alpha$, the mean value theorem gives $|P(p/q)|\le M|p/q-\alpha |$, where $M=\max |P^{\prime }|>0$ on that interval. Thus rational points in the interval satisfy $|\alpha -p/q|\ge M^{-1}{q}^{-d}$. Outside it, a fixed positive distance from $\alpha$ gives the same type of bound after decreasing the constant. Choosing a smaller constant makes the inequality strict, proving the Liouville approximation theorem.

For nonreal algebraic numbers, the positive distance from the real axis already gives a uniform lower bound against rational approximations. The substantive approximation argument is therefore the real case.