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In [[mathematics]], '''Stickelberger's theorem''' is a result of [[algebraic number theory]], which gives some information about the [[Galois module]] structure of [[class group]]s of [[cyclotomic field]]s. A special case was first proven by [[Ernst Kummer]] ([[#{{harvid|Kummer|1847}}|1847]]) while the general result is due to [[Ludwig Stickelberger]] ([[#{{harvid|Stickelberger|1890}}|1890]]).<ref>{{harvnb|Washington|1997|loc=Notes to chapter 6}}</ref>
 
==The Stickelberger element and the Stickelberger ideal==
Let ''K<sub>m</sub>'' denote the ''m''th [[cyclotomic field]], i.e. the [[field extension|extension]] of the [[rational number]]s obtained by [[Adjunction (field theory)|adjoining]] the ''m''th [[root of unity|roots of unity]] to '''Q''' (where ''m''&nbsp;≥&nbsp;2 is an integer). It is a [[Galois extension]] of '''Q''' with [[Galois group]] ''G<sub>m</sub>'' isomorphic to the [[Multiplicative group of integers modulo n|multiplicative group of integers modulo ''m'']] ('''Z'''/''m'''''Z''')<sup>×</sup>. The '''Stickelberger element''' ('''of level ''m''''' or '''of''' ''K<sub>m</sub>'') is an element in the [[group ring]] '''Q'''[''G<sub>m</sub>''] and the '''Stickelberger ideal''' ('''of level ''m''''' or '''of''' ''K<sub>m</sub>'') is an ideal in the group ring '''Z'''[''G<sub>m</sub>'']. They are defined as follows. Let ζ<sub>''m''</sub> denote a [[primitive root of unity|primitive ''m''th root of unity]]. The isomorphism from ('''Z'''/''m'''''Z''')<sup>×</sup> to ''G<sub>m</sub>'' is given by sending ''a'' to σ<sub>''a''</sub> defined by the relation
:σ<sub>''a''</sub>(ζ)&nbsp;=&nbsp;{{SubSup|ζ|''m''|''a''}}.
The Stickelberger element of level ''m'' is defined as
:<math>\theta(K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\sigma_a^{-1}\in\mathbf{Q}[G_m].</math>
The Stickelberger ideal of level ''m'', denoted ''I''(''K<sub>m</sub>''), is the set of integral multiples of θ(''K<sub>m</sub>'') which have integral coefficients, i.e.
:<math>I(K_m)=\theta(K_m)\mathbf{Z}[G_m]\cap\mathbf{Z}[G_m].</math>
 
More generally, if ''F'' be any [[abelian number field]] whose Galois group over '''Q''' is denoted ''G<sub>F</sub>'', then the '''Stickelberger element of''' ''F'' and the '''Stickelberger ideal of''' ''F'' can be defined. By the [[Kronecker–Weber theorem]] there is an integer ''m'' such that ''F'' is contained in ''K<sub>m</sub>''. Fix the least such ''m'' (this is the (finite part of the) [[conductor (class field theory)|conductor]] of ''F'' over '''Q'''). There is a natural [[group homomorphism]] ''G<sub>m</sub>''&nbsp;→&nbsp;''G<sub>F</sub>'' given by restriction, i.e. if σ&nbsp;∈&nbsp;''G<sub>m</sub>'', its image in ''G<sub>F</sub>'' is its restriction to ''F'' denoted res<sub>''m''</sub>σ. The Stickelberger element of ''F'' is then defined as
:<math>\theta(F)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\mathrm{res}_m\sigma_a^{-1}\in\mathbf{Q}[G_F].</math>
The Stickelberger ideal of ''F'', denoted ''I''(''F''), is defined as in the case of ''K<sub>m</sub>'', i.e.
:<math>I(F)=\theta(F)\mathbf{Z}[G_F]\cap\mathbf{Z}[G_F].</math>
 
In the special case where ''F''&nbsp;=&nbsp;''K<sub>m</sub>'', the Stickelberger ideal ''I''(''K<sub>m</sub>'') is generated by (''a''&nbsp;&minus;&nbsp;σ<sub>''a''</sub>)θ(''K<sub>m</sub>'') as ''a'' varies over '''Z'''/''m'''''Z'''. This not true for general ''F''.<ref>{{harvnb|Washington|1997}}, Lemma 6.9 and the comments following it</ref>
 
===Examples===
*If ''F'' is a [[totally real field]] of conductor ''m'', then<ref>{{harvnb|Washington|1997|loc=§6.2}}</ref>
::<math>\theta(F)=\frac{\phi(m)}{2[F:\mathbf{Q}]}\sum_{\sigma\in G_F}\sigma,</math>
:where φ is the [[Euler totient function]] and [''F''&nbsp;:&nbsp;'''Q'''] is the [[degree of a field extension|degree]] of ''F'' over '''Q'''.
 
==Statement of the theorem==
:'''Stickelberger's Theorem'''<ref>{{harvnb|Washington|1997|loc=Theorem 6.10}}</ref>
:Let ''F'' be an abelian number field. Then, the Stickelberger ideal of ''F'' [[Annihilator (ring theory)|annihilates]] the class group of ''F''.
 
Note that θ(''F'') itself need not be an annihilator, but any multiple of it in '''Z'''[''G<sub>F</sub>''] is.
 
Explicitly, the theorem is saying that if α&nbsp;∈&nbsp;'''Z'''[''G<sub>F</sub>''] is such that
:<math>\alpha\theta(F)=\sum_{\sigma\in G_F}a_\sigma\sigma\in\mathbf{Z}[G_F]</math>
and if ''J'' is any [[fractional ideal]] of ''F'', then
:<math>\prod_{\sigma\in G_F}\sigma(J^{a_\sigma})</math>
is a [[principal ideal]].
 
== See also ==
 
*[[Gross–Koblitz formula]]
*[[Herbrand–Ribet theorem]]
 
== Notes ==
{{reflist}}
 
== References ==
* {{cite book |last=Cohen |first=Henri |authorlink= Henri Cohen (number theorist) | year=2007 | title=Number Theory – Volume I: Tools and Diophantine Equations | isbn= 978-0-387-49922-2 | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=239| zbl=1119.11001 | pages=150–170 }}
*Boas Erez, [http://www.fen.bilkent.edu.tr/~franz/publ/boas.pdf ''Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung'']
* {{cite book | zbl=0376.12002 | last=Fröhlich | first=A. | author-link=Albrecht Fröhlich | chapter=Stickelberger without Gauss sums | pages=589-607 | title=Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 | editor1-last=Fröhlich | editor1-first=A. | editor1-link=Albrecht Fröhlich | publisher=Academic Press | year=1977 | isbn=0-12-268960-7 | zbl=0359.12015 }}
*{{Citation
| last=Kummer
| first=Ernst
| author-link=Ernst Kummer
| title=Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren
| year=1847
| journal=Journal für die Reine und Angewandte Mathematik
| volume=35
| pages=327–367
}}
*{{Citation
| last=Stickelberger
| first=Ludwig
| author-link=Ludwig Stickelberger
| title=Ueber eine Verallgemeinerung der Kreistheilung
| url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27547
| journal=Mathematische Annalen
| volume=37
| number=3
| year=1890
| pages=321–367
| mr=1510649 | jfm = 22.0100.01
}}
*{{Citation
| last=Washington
| first=Lawrence
| title=Introduction to Cyclotomic Fields
| edition=2
| publisher=Springer-Verlag
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-0-387-94762-4
| mr=1421575
| year=1997
| volume=83
}}
 
== External links ==
*[http://planetmath.org/?op=getobj&from=objects&id=5642 PlanetMath page]
 
[[Category:Cyclotomic fields]]
[[Category:Theorems in number theory]]

Latest revision as of 15:41, 8 January 2014

In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer ([[#Template:Harvid|1847]]) while the general result is due to Ludwig Stickelberger ([[#Template:Harvid|1890]]).[1]

The Stickelberger element and the Stickelberger ideal

Let Km denote the mth cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the mth roots of unity to Q (where m ≥ 2 is an integer). It is a Galois extension of Q with Galois group Gm isomorphic to the multiplicative group of integers modulo m (Z/mZ)×. The Stickelberger element (of level m or of Km) is an element in the group ring Q[Gm] and the Stickelberger ideal (of level m or of Km) is an ideal in the group ring Z[Gm]. They are defined as follows. Let ζm denote a primitive mth root of unity. The isomorphism from (Z/mZ)× to Gm is given by sending a to σa defined by the relation

σa(ζ) = Template:SubSup.

The Stickelberger element of level m is defined as

θ(Km)=1ma=1m(a,m)=1aσa1Q[Gm].

The Stickelberger ideal of level m, denoted I(Km), is the set of integral multiples of θ(Km) which have integral coefficients, i.e.

I(Km)=θ(Km)Z[Gm]Z[Gm].

More generally, if F be any abelian number field whose Galois group over Q is denoted GF, then the Stickelberger element of F and the Stickelberger ideal of F can be defined. By the Kronecker–Weber theorem there is an integer m such that F is contained in Km. Fix the least such m (this is the (finite part of the) conductor of F over Q). There is a natural group homomorphism Gm → GF given by restriction, i.e. if σ ∈ Gm, its image in GF is its restriction to F denoted resmσ. The Stickelberger element of F is then defined as

θ(F)=1ma=1m(a,m)=1aresmσa1Q[GF].

The Stickelberger ideal of F, denoted I(F), is defined as in the case of Km, i.e.

I(F)=θ(F)Z[GF]Z[GF].

In the special case where F = Km, the Stickelberger ideal I(Km) is generated by (a − σa)θ(Km) as a varies over Z/mZ. This not true for general F.[2]

Examples

θ(F)=ϕ(m)2[F:Q]σGFσ,
where φ is the Euler totient function and [F : Q] is the degree of F over Q.

Statement of the theorem

Stickelberger's Theorem[4]
Let F be an abelian number field. Then, the Stickelberger ideal of F annihilates the class group of F.

Note that θ(F) itself need not be an annihilator, but any multiple of it in Z[GF] is.

Explicitly, the theorem is saying that if α ∈ Z[GF] is such that

αθ(F)=σGFaσσZ[GF]

and if J is any fractional ideal of F, then

σGFσ(Jaσ)

is a principal ideal.

See also

Notes

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  • Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
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External links

  1. Template:Harvnb
  2. Template:Harvnb, Lemma 6.9 and the comments following it
  3. Template:Harvnb
  4. Template:Harvnb