Solution of Bachet Diophantine equation with algebraic number theory

Keywords: Algebraic numbers, Algebraic number fields, Dedekind domains, Ring of algebraic integers, Fractional ideals, Fundamental unit, Class number, Bachet equation This study consists of five parts. In the first part, some definitions and theorems related to rings, ideals, integral domains and Noetherian domains are given. In the second part, algebraic numbers, algebraic number fields and algebraic field extensions are examined. In the third part, definitions of Dedekind domain, fractional ideal, fundamental unit and class number are given. The greatest common divisor and the least common multiple of two ideals are given. Later, some theorems concerning the greatest common divisor and the least common multiple are proved. In the fourth part, the solutions of the Bachet equations are examined. And if they have solutions, these solutions are given. The fifth part consists of conclusions and recomendations.