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 Z_''p''-extensions of abelian algebraic number fields.

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'' we let ''K''_''m'' denote the extension by ''p''^''m''-power roots of unity, $\hat K$ the union of the ''K''_''m'' and ''A''^(''p'') the maximal unramified abelian ''p''-extension of $\hat K$. Let the Tate module
:$T_p(K) = \mathrm(A^/\hat K) \ .$
Then ''T''_''p''(''K'') is a pro-''p''-group and so a Z_''p''-module. Using class field theory one can describe ''T''_''p''(''K'') as isomorphic to the inverse limit of the class groups ''C''_''m'' of the ''K''_''m'' under norm.

Iwasawa exhibited ''T''_''p''(''K'') 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
:$\lambda m + \mu p^m + \kappa \ .$
The Ferrero–Washington theorem states that μ is zero.

References

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{{DEFAULTSORT:Ferrero-Washington theorem
Theorems in algebraic number theory