Ferrero–Washington theorem

In algebraic number theory, the Ferrero–Washington theorem states that Iwasawa's μ-invariant vanishes for cyclotomic Z_p-extensions of abelian algebraic number fields. It was first proved by. A different proof was given by.

History

introduced the μ-invariant of a Z_p-extension and observed that it was zero in all cases he calculated. used a computer to check that it vanishes for the cyclotomic Z_p-extension of the rationals for all primes less than 4000.
later conjectured that the μ-invariant vanishes for any Z_p-extension, but shortly after discovered examples of non-cyclotomic extensions of number fields with non-vanishing μ-invariant showing that his original conjecture was wrong. He suggested, however, that the conjecture might still hold for cyclotomic Z_p-extensions.
showed that the vanishing of the μ-invariant for cyclotomic Z_p-extensions of the rationals is equivalent to certain congruences between Bernoulli numbers, and showed that the μ-invariant vanishes in these cases by proving that these congruences hold.

Statement

For a number field K, denote the extension of K by p^m-power roots of unity by K_m, the union of the K_m as m ranges over all positive integers by, and the maximal unramified abelian p-extension of by A. Let the Tate module
Then T_p is a pro-p-group and so a Z_p-module. Using class field theory one can describe T_p as isomorphic to the inverse limit of the class groups C_m of the K_m under norm.
Iwasawa exhibited T_p as a module over the completion Z_p and this implies a formula for the exponent of p in the order of the class groups C_m of the form
The Ferrero–Washington theorem states that μ is zero.