Ferrero–Washington theorem
Template:Short description Template:Infobox mathematical statement In algebraic number theory, the Ferrero–Washington theorem states that Iwasawa's μ-invariant vanishes for cyclotomic Zp-extensions of abelian algebraic number fields. It was first proved by Template:Harvtxt. A different proof was given by Template:Harvtxt.
History
Template:Harvtxt introduced the μ-invariant of a Zp-extension and observed that it was zero in all cases he calculated. Template:Harvtxt used a computer to check that it vanishes for the cyclotomic Zp-extension of the rationals for all primes less than 4000. Template:Harvtxt later conjectured that the μ-invariant vanishes for any Zp-extension, but shortly after Template:Harvtxt 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.
Template:Harvtxt showed that the vanishing of the μ-invariant for cyclotomic Zp-extensions of the rationals is equivalent to certain congruences between Bernoulli numbers, and Template:Harvtxt 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 pm-power roots of unity by Km, the union of the Km as m ranges over all positive integers by , and the maximal unramified abelian p-extension of by A(p). Let the Tate module
Then Tp(K) is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp(K) as isomorphic to the inverse limit of the class groups Cm of the Km under norm.[1]
Iwasawa exhibited Tp(K) as a module over the completion ZpTemplate:Brackets 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.[2]