proof of Liouville approximation theorem


Let α satisfy the equation f(α)=anαn+an-1αn-1++a0=0 where the ai are integers. Choose M such that M>maxα-1xα+1|f(x)|.

Suppose pq lies in (α-1,α+1) and f(pq)0.

|f(pq)|=|anpn+an-1pn-1q++a0qn|qn1qn

since the numerator is a non-zero integer.

1qn|f(pq)-f(α)|=|(pq-α)f(x)|<M|pq-α|.
Title proof of Liouville approximation theorem
Canonical name ProofOfLiouvilleApproximationTheorem
Date of creation 2013-03-22 13:19:22
Last modified on 2013-03-22 13:19:22
Owner lieven (1075)
Last modified by lieven (1075)
Numerical id 6
Author lieven (1075)
Entry type Proof
Classification msc 11J68