How Can We Construct Abelian Galois Extensions of Basic Number Fields?

Abstract

Irregular primes-37 being the first such prime-have played a great role in number theory. This article discusses Ken Ribet's construction-for all irregular primes p-of specific abelian, unramified, degree p extensions of the number fields Q(e(2 pi i/p)). These extensions with explicit information about their Galois groups (they are Galois over Q) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet's method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.