Liouville’s Theorem

Abstract

A complex number α is said to be an algebraic number if there is a non-zero polynomial $f(x) \in \mathbb{Q}[x]$ such that f(α) = 0. Given an algebraic number α, there exists a unique irreducible monic polynomial $P(x) \in \mathbb{Q}[x]$ such that P(α) = 0. This is called the minimal polynomial of α. The set of all algebraic numbers denoted by $\overline{\mathbb{Q}}$ is a subfield of the field of complex numbers. A complex number which is not algebraic is said to be transcendental.