General remarks

Dedekind's invention of ideals in the 1870s was a major turning point in the development of algebra. His aim was to apply ideals to number theory, but to do this he had to build the whole framework of commutative algebra: fields, rings, modules and vector spaces. These concepts, together with groups, were to form the core of the future abstract algebra. At the same time, he created algebraic number theory, which became the temporary home of algebra while its core concepts were growing up. Algebra finally became independent in the 1920s, when fields, rings and modules were generalised beyond the realm of numbers by Emmy Noether and Emil Artin. But even then, Emmy Noether used to say “Es steht schon bei Dedekind” (“It's already in Dedekind”), and urged her students to read all of Dedekind's works in ideal theory.

Today this is still worthwhile, but not so easy. Dedekind wrote for an audience that knew number theory – especially quadratic forms – but not the concepts of ring, field or module. Today's readers probably have the opposite qualifications, and of course most are not fluent in German and French. In an attempt to overcome these problems, I have translated the most accessible of Dedekind's works on ideal theory, Sur la Théorie des Nombres Entiers Algébriques, Dedekind (1877), which he wrote to explain his ideas to a general mathematical audience. This memoir shows the need for ideals in a very concrete case, the numbers m + n√–5 where m, n ∈ ℤ, before going on to develop a general theory and to prove the theorem on unique factorisation into prime ideals.