Cyclotomic Fields

Kummer's exploration of cyclotomic fields significantly influenced the evolution of algebraic number theory, laying the groundwork for later mathematicians like Dedekind and Hilbert. Despite its foundational role, Kummer's specific contributions were overshadowed until the mid-20th century when Iwasawa and Leopoldt revived interest in cyclotomic fields. Iwasawa's work focused on their analogues in algebraic geometry, while Leopoldt developed p-adic analogues of classical formulas, ultimately revealing a vital link between cyclotomic towers and p-adic L-functions.