Liouville’s theorem on diophantine approximation

Liouville’s theorem on diophantine approximation Let be an irrational number that is algebraic of degree . Then for any real number , there are at most finitely many rational numbers such that .

Proof. If there is no such rational number then the number of solution is clearly finite.

Now assume there exists at least one rational number such that .

Let be a polynomial of degree , with integers coefficients, such that . Choose such that has no roots other that on the interval

Write where is a monic polynomial with real coefficients of degree . Since is continuous, there exists such that on the interval

For all rational number such that , or

1. For all rational number such that . and , it is not difficult to tell the number of such rational numbers will be finite.

2. For all rational number such that . Note

   Since has degree and integer coefficients,

   where both the numerator and the denominator are integers. Since is irrational, . Then . Note

   Similarly, we can deduce the number of rational numbers such that will be finite.

Therefore, there are at most finitely many rational numbers such that .

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