Ferrero–Washington theorem


In algebraic number theory, the Ferrero–Washington theorem, proved first by and later by, states that Iwasawa's μ-invariant vanishes for cyclotomic Zp-extensions of abelian algebraic number fields.

History

introduced the μ-invariant of a Zp-extension and observed that it was zero in all cases he calculated. used a computer to check that it vanishes for the cyclotomic Zp-extension of the rationals for all primes less than 4000.
later conjectured that the μ-invariant vanishes for any Zp-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 Zp-extensions.
showed that the vanishing of the μ-invariant for cyclotomic Zp-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 we let Km denote the extension by pm-power roots of unity, the union of the Km and A the maximal unramified abelian p-extension of. Let the Tate module
Then Tp is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp as isomorphic to the inverse limit of the class groups Cm of the Km under norm.
Iwasawa exhibited Tp as a module over the completion Zp and this implies a formula for the exponent of p in the order of the class groups Cm of the form
The Ferrero–Washington theorem states that μ is zero.