Arithmetic of Algebraic Numbers

Abstract

The idea to extend the field of rational numbers owes a lot to various attempts to solve some concrete Diophantine equations. The use of irrational numbers which are roots of polynomials with rational coefficients often makes it possible to reduce such equations to more convenient forms. An intriguing example of this is the study of the Fermat equation (Borevich Z.I., Shafarevich I.R. (1985), Postnikov M.M. (1978), Edwards H.M. (1977), Ribenboim P. (1988))

$$ {x^n} + {y^n} = {z^n}\quad \left( {n>2} \right) $$

((1.1))

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