by a cyclic extension of degree p which behaves in the specified way under the action of Σ; Ribet proves this by actually constructing such an extension using methods in the theory of modular forms. A more elementary proof of Ribet's converse to Herbrand's theorem, a consequence of the theory of Euler systems, can be found in Washington's book. Generalizations Ribet's methods were developed further by Barry Mazur and Andrew Wiles in order to prove the main conjecture of Iwasawa theory, a corollary of which is a strengthening of the Herbrand–Ribet theorem: the power of p dividing Bp−n is exactly the power of p dividing the order of Gn. See also Iwasawa theory Notes Cyclotomic fields Theorems in algebraic number theory