CHAPTER VIII - RING OF RATIONAL POLYNOMIALS. ALGEBRAIC AND TRANSCENDENTAL NUMBERS

Abstract

This chapter discusses the algebraic and transcendental numbers of rational polynomials. It presents a theorem which states that if a polynomial with integral coefficients can be expressed as a product of two polynomials with rational coefficients, and then it can be expressed also as a product of two polynomials with integral coefficients. Algebraic numbers form a field. A root of a polynomial with algebraic coefficients is an algebraic number. The chapter also discusses transcendental numbers. A complex number that is not algebraic is called transcendental. The existence of transcendental numbers was proved first by Liouville in 1851. Much more difficult than the proof of existence of transcendental numbers is the investigation as to whether given number is transcendental or algebraic.