Algebraic Number Theory: Cyclotomy

Abstract

In this chapter, we return to one of Gauss’s favourite themes, cyclotomic integers, and look at how they were used by Kummer, one of the leaders of the next generation of German number theorists. French and German mathematicians did not keep up-to-date with each other’s work, and for a brief, exciting moment in Paris in 1847 it looked as if the cyclotomic integers offered a chance to prove Fermat’s last theorem, only for Kummer to report, via Liouville, that problems with the concept of a prime cyclotomic integer wrecked that hope. Primality, however, turned out to be a much more interesting concept, and one of the roots of the concept of an ideal.