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<p><b>New page</b></p><div>In algebraic number theory, the '''Ferrero–Washington theorem''', proved first by {{harvtxt|Ferrero|Washington|1979}} and later by {{harvtxt|Sinnott|1984}}, states that [[Iwasawa's μ-invariant]] vanishes for cyclotomic '''Z'''<sub>''p''</sub>-extensions of abelian [[algebraic number field]]s.<br />
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==History==<br />
{{harvtxt|Iwasawa|1959}} introduced the μ-invariant of a '''Z'''<sub>''p''</sub>-extension and observed that it was zero in all cases he calculated. {{harvtxt|Iwasawa|Sims|1966}} used a computer to check that it vanishes for the cyclotomic '''Z'''<sub>''p''</sub>-extension of the rationals for all [[prime number|primes]] less than 4000. <br />
{{harvtxt|Iwasawa|1971}} later conjectured that the μ-invariant vanishes for any '''Z'''<sub>''p''</sub>-extension, but shortly after {{harvtxt|Iwasawa|1973}} 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'''<sub>''p''</sub>-extensions.<br />
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{{harvtxt|Iwasawa|1958}} showed that the vanishing of the μ-invariant for cyclotomic '''Z'''<sub>''p''</sub>-extensions of the rationals is equivalent to certain congruences between [[Bernoulli number]]s, and {{harvtxt|Ferrero|Washington|1979}} showed that the μ-invariant vanishes in these cases by proving that these congruences hold.<br />
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==Statement==<br />
For a number field ''K'' we let ''K''<sub>''m''</sub> denote the extension by ''p''<sup>''m''</sup>-power roots of unity, <math>\hat K</math> the union of the ''K''<sub>''m''</sub> and ''A''<sup>(''p'')</sup> the maximal unramified abelian ''p''-extension of <math>\hat K</math>. Let the [[Tate module of a number field|Tate module]] <br />
:<math>T_p(K) = \mathrm{Gal}(A^{(p)}/\hat K) \ . </math><br />
Then ''T''<sub>''p''</sub>(''K'') is a pro-''p''-group and so a '''Z'''<sub>''p''</sub>-module. Using [[class field theory]] one can describe ''T''<sub>''p''</sub>(''K'') as isomorphic to the inverse limit of the class groups ''C''<sub>''m''</sub> of the ''K''<sub>''m''</sub> under norm.<ref name=MP245>{{harvnb|Manin|Panchishkin|2007|p=245}}</ref><br />
<br />
Iwasawa exhibited ''T''<sub>''p''</sub>(''K'') as a module over the completion '''Z'''<sub>''p''</sub>[<span/>[''T'']] and this implies a formula for the exponent of ''p'' in the order of the class groups ''C''<sub>''m''</sub> of the form<br />
:<math> \lambda m + \mu p^m + \kappa \ . </math><br />
The Ferrero–Washington theorem states that μ is zero.<ref name=MP246>{{harvnb|Manin|Panchishkin|2007|p=246}}</ref><br />
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==References==<br />
{{reflist}}<br />
*{{Citation | author1-link=Bruce Ferrero | author2-link=Larry Washington | last1=Ferrero | first1=Bruce | last2=Washington | first2=Lawrence C. | title=The Iwasawa invariant μ<sub>p</sub> vanishes for abelian number fields | url=http://dx.doi.org/10.2307/1971116 | doi=10.2307/1971116 | year=1979 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=109 | issue=2 | pages=377–395 | mr=528968 | zbl=0443.12001 }}<br />
*{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=On some invariants of cyclotomic fields | url=http://www.jstor.org/stable/2372782 | id={{MR|0124317}}[http://www.jstor.org/stable/2372857 correction] | year=1958 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=81 | pages=280}}<br />
*{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=On Γ-extensions of algebraic number fields | doi=10.1090/S0002-9904-1959-10317-7 | id={{MR|0124316}} | year=1959 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=65 | issue=4 | pages=183–226}}<br />
*{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 | url=http://ada00.math.uni-bielefeld.de/ICM/ICM1970.1/ | publisher=Gauthier-Villars | id={{MR|0422205}} | year=1971 | chapter=On some infinite Abelian extensions of algebraic number fields | pages=391–394}}<br />
*{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki | url=http://books.google.com/books?id=4_buAAAAMAAJ | publisher=Kinokuniya | location=Tokyo | id={{MR|0357371}} | year=1973 | chapter=On the μ-invariants of Z1-extensions | pages=1–11}}<br />
*{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | last2=Sims | first2=Charles C. | title=Computation of invariants in the theory of cyclotomic fields | doi=10.4099/jmath.18.86 | id={{MR|0202700}} | year=1966 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=18 | pages=86–96}}<br />
*{{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 }}<br />
*{{Citation | last1=Sinnott | first1=W. | title=On the μ-invariant of the Γ-transform of a rational function | url=http://dx.doi.org/10.1007/BF01388565 | doi=10.1007/BF01388565 | year=1984 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=75 | issue=2 | pages=273–282 | mr=732547 | zbl=0531.12004 }}<br />
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{{DEFAULTSORT:Ferrero-Washington theorem}}<br />
[[Category:Algebraic number theory]]<br />
[[Category:Theorems in algebraic number theory]]</div>en>Rjwilmsi