Liouville’s Theorem

M. Ram Murty and Purusottam Rath
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
Purusottam Rath: Chennai Mathematical Institute

Chapter Chapter 1 in Transcendental Numbers, 2014, pp 1-6 from Springer

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

Keywords: Complex Algebraic Numbers; Minimal Polynomial; Liouville Number; Algebraic Independence; Diophantine Applications (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-1-4939-0832-5_1

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