algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, the Ferrero–Washington theorem states that

Iwasawa's μ-invariant In mathematics, the Iwasawa algebra Λ(''G'') of a profinite group ''G'' is a variation of the group ring of ''G'' with ''p''-adic coefficients that take the topology of ''G'' into account. More precisely, Λ(''G'') is the inverse limit of the gro ...

vanishes for cyclotomic Z_''p''-extensions of abelian

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s. 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 number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...

s, 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

\hat K

, and the maximal unramified abelian ''p''-extension of

\hat K

by ''A''^(''p''). Let the

Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', '' ...

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 In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

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

Sources

* * (And correction ) * * * * * * {{DEFAULTSORT:Ferrero-Washington theorem Theorems in algebraic number theory