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(e2πi/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. © 2011 American Mathematical Society Reverts to public domain 28 years from publication. © 2011 American Mathematical Society.