Number theory Diophantine approximation and transcendence Theorem

Liouville's theorem on Diophantine approximation

Builds on Liouville number Algebraic number Transcendental number

Statement

Let $\alpha$ be a real algebraic number of degree $d \ge 2$ . Then there is a constant $c > 0$ such that $\left| \alpha - \frac{p}{q} \right| > \frac{c}{q^d}$ for all integers $p$ and $q$ with $q > 0$ . As a consequence, each Liouville number is transcendental.

Remarks

The proof uses the minimal polynomial $f$ of $\alpha$ : $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\alpha - p/q|$ . For a Liouville number $x$ , the approximations are better than $c / q^d$ for every $d$ , so $x$ is not algebraic. Roth's theorem improves the exponent $d$ to $2 + \varepsilon$ for each $\varepsilon > 0$ .

History

Joseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Algebraic number definition
Liouville number definition
Transcendental number definition
Liouville's theorem on Diophantine approximation statement

Tag	`0N3WFZEFPM`
URI	`m2://0N3WFZEFPM`
Kind	statement, theorem
Writing	W0 stub
MSC	11J68
Status	This copy of the site has no build data. Run `m2 status 0N3WFZEFPM`.
Uses	Liouville number, Algebraic number, Transcendental number

Liouville's theorem on Diophantine approximation

Statement

Remarks

History

Reading path

Further reading

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