Stickelberger's theorem

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 (1847) while the general result is due to Ludwig Stickelberger (1890).^[1]

The Stickelberger element and the Stickelberger ideal

Let $K m$ denote the $m$ th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the $m$ th roots of unity to $\mathbb {Q}$ (where $m \geq 2$ is an integer). It is a Galois extension of $\mathbb {Q}$ with Galois group $G m$ isomorphic to the multiplicative group of integers modulo $m$ $\mathbb {Z}$ . The Stickelberger element (of level $m$ or of $K m$ ) is an element in the group ring $\mathbb {Q}$ and the Stickelberger ideal (of level $m$ or of $K m$ ) is an ideal in the group ring $\mathbb {Z}$ . They are defined as follows. Let $ζ m$ denote a primitive $m$ th root of unity. The isomorphism from $\mathbb {Z}$ to $G m$ is given by sending $a$ to $σ a$ defined by the relation

\sigma _{a}(\zeta _{m})=\zeta _{m}^{a}

.

The Stickelberger element of level $m$ is defined as

\theta (K_{m})={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \sigma _{a}^{-1}\in \mathbb {Q} [G_{m}].

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

I(K_{m})=\theta (K_{m})\mathbb {Z} [G_{m}]\cap \mathbb {Z} [G_{m}].

More generally, if $F$ be any Abelian number field whose Galois group over $\mathbb {Q}$ is denoted $G F$ , 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 $K m$ . Fix the least such $m$ (this is the (finite part of the) conductor of $F$ over $\mathbb {Q}$ ). There is a natural group homomorphism $G m \to G F$ given by restriction, i.e. if $σ \in G m$ , its image in $G F$ is its restriction to $F$ denoted $res m σ$ . The Stickelberger element of $F$ is then defined as

\theta (F)={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \mathrm {res} _{m}\sigma _{a}^{-1}\in \mathbb {Q} [G_{F}].

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

I(F)=\theta (F)\mathbb {Z} [G_{F}]\cap \mathbb {Z} [G_{F}].

In the special case where $F = K m$ , the Stickelberger ideal $I (K m)$ is generated by $(a - σ a) θ (K m)$ as $a$ varies over $\mathbb {Z}$ . This not true for general F.^[2]

Examples

If $F$ is a totally real field of conductor $m$ , then^[3]

\theta (F)={\frac {\varphi (m)}{2[F:\mathbb {Q} ]}}\sum _{\sigma \in G_{F}}\sigma ,

where $φ$ is the Euler totient function and $\mathbb {Q}$ is the degree of $F$ over $\mathbb {Q}$ .

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 $\mathbb {Z}$ is.

Explicitly, the theorem is saying that if $\mathbb {Z}$ is such that

\alpha \theta (F)=\sum _{\sigma \in G_{F}}a_{\sigma }\sigma \in \mathbb {Z} [G_{F}]

and if $J$ is any fractional ideal of $F$ , then

\prod _{\sigma \in G_{F}}\sigma \left(J^{a_{\sigma }}\right)

is a principal ideal.

Notes

^ Washington 1997, Notes to chapter 6
^ Washington 1997, Lemma 6.9 and the comments following it
^ Washington 1997, §6.2
^ Washington 1997, Theorem 6.10

References

Cohen, Henri (2007). Number Theory – Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239. Springer-Verlag. pp. 150–170. ISBN 978-0-387-49922-2. Zbl 1119.11001.
Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
Fröhlich, A. (1977). "Stickelberger without Gauss sums". In Fröhlich, A. (ed.). Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975. Academic Press. pp. 589–607. ISBN 0-12-268960-7. Zbl 0376.12002.
Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Vol. 84 (2nd ed.). New York: Springer-Verlag. doi:10.1007/978-1-4757-2103-4. ISBN 978-1-4419-3094-1. MR 1070716.
Kummer, Ernst (1847), "Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren", Journal für die Reine und Angewandte Mathematik, 1847 (35): 327–367, doi:10.1515/crll.1847.35.327, S2CID 123230326
Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen, 37 (3): 321–367, doi:10.1007/bf01721360, JFM 22.0100.01, MR 1510649, S2CID 121239748
Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575

External links

PlanetMath page

[1] Washington 1997, Notes to chapter 6

[2] Washington 1997, Lemma 6.9 and the comments following it

[3] Washington 1997, §6.2

[4] Washington 1997, Theorem 6.10

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