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]

Let ${\displaystyle K_{m}}$ denote the ${\displaystyle m}$th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the ${\displaystyle m}$th roots of unity to ${\displaystyle \mathbb {Q} }$ (where ${\displaystyle m\geq 2}$ is an integer). It is a Galois extension of ${\displaystyle \mathbb {Q} }$ with Galois group ${\displaystyle G_{m}}$ isomorphic to the multiplicative group of integers modulo m ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$. The Stickelberger element (of level ${\displaystyle m}$ or of ${\displaystyle K_{m}}$) is an element in the group ring ${\displaystyle \mathbb {Q} [G_{m}]}$ and the Stickelberger ideal (of level ${\displaystyle m}$ or of ${\displaystyle K_{m}}$) is an ideal in the group ring ${\displaystyle \mathbb {Z} [G_{m}]}$. They are defined as follows. Let ${\displaystyle \zeta _{m}}$ denote a primitive ${\displaystyle m}$th root of unity. The isomorphism from ${\displaystyle (\mathbb {Z} /m\mathbb {Z} )^{\times }}$ to ${\displaystyle G_{m}}$ is given by sending an element ${\displaystyle a}$ to ${\displaystyle \sigma _{a}}$ defined by the relation $${\displaystyle \sigma _{a}(\zeta _{m})=\zeta _{m}^{a}.}$$ The Stickelberger element of level ${\displaystyle m}$ is defined as $${\displaystyle \theta (K_{m})={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \sigma _{a}^{-1}\in \mathbb {Q} [G_{m}].}$$ The Stickelberger ideal of level ${\displaystyle m}$, denoted ${\displaystyle I(K_{m})}$, is the set of integral multiples of ${\displaystyle \theta (K_{m})}$ which have integral coefficients, i.e. $${\displaystyle I(K_{m})=\theta (K_{m})\mathbb {Z} [G_{m}]\cap \mathbb {Z} [G_{m}].}$$

More generally, if ${\displaystyle F}$ be any Abelian number field whose Galois group over ${\displaystyle \mathbb {Q} }$ is denoted ${\displaystyle G_{F}}$, then the Stickelberger element of ${\displaystyle F}$ and the Stickelberger ideal of ${\displaystyle F}$ can be defined. By the Kronecker–Weber theorem there is an integer ${\displaystyle m}$ such that ${\displaystyle F}$ is contained in ${\displaystyle K_{m}}$. Fix the least such ${\displaystyle m}$ (this is the (finite part of the) conductor of ${\displaystyle F}$ over ${\displaystyle \mathbb {Q} }$). There is a natural group homomorphism ${\displaystyle G_{m}\to G_{F}}$ given by restriction, i.e. if ${\displaystyle \sigma _{\in }G_{m}}$, its image in ${\displaystyle G_{F}}$ is its restriction to ${\displaystyle F}$ denoted ${\displaystyle \operatorname {res} _{m}\sigma }$. The Stickelberger element of ${\displaystyle F}$ is then defined as $${\displaystyle \theta (F)={\frac {1}{m}}{\underset {(a,m)=1}{\sum _{a=1}^{m}}}a\cdot \mathrm {res} _{m}\sigma _{a}^{-1}\in \mathbb {Q} [G_{F}].}$$ The Stickelberger ideal of ${\displaystyle F}$, denoted ${\displaystyle I(F)}$, is defined as in the case of ${\displaystyle K_{m}}$, i.e. $${\displaystyle I(F)=\theta (F)\mathbb {Z} [G_{F}]\cap \mathbb {Z} [G_{F}].}$$

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

If ${\displaystyle F}$ is a totally real field of conductor ${\displaystyle m}$, then^[3] $${\displaystyle \theta (F)={\frac {\varphi (m)}{2[F:\mathbb {Q} ]}}\sum _{\sigma \in G_{F}}\sigma ,}$$ where ${\displaystyle \varphi }$ is the Euler totient function and ${\displaystyle [F:\mathbb {Q} ]}$ is the degree of ${\displaystyle F}$ over ${\displaystyle \mathbb {Q} }$.

Stickelberger's Theorem^[4]
Let ${\displaystyle F}$ be an abelian number field. Then, the Stickelberger ideal of ${\displaystyle F}$ annihilates the class group of ${\displaystyle F}$.

Note that ${\displaystyle \theta (F)}$ itself need not be an annihilator, but any multiple of it in ${\displaystyle \mathbb {Z} [G_{F}]}$ is.

Explicitly, the theorem is saying that if ${\displaystyle \alpha \in \mathbb {Z} [G_{F}]}$ is such that $${\displaystyle \alpha \theta (F)=\sum _{\sigma \in G_{F}}a_{\sigma }\sigma \in \mathbb {Z} [G_{F}]}$$ and if ${\displaystyle J}$ is any fractional ideal of ${\displaystyle F}$, then $${\displaystyle \prod _{\sigma \in G_{F}}\sigma \left(J^{a_{\sigma }}\right)}$$ is a principal ideal.

• Gross–Koblitz formula
• Herbrand–Ribet theorem
• Thaine's theorem
• Jacobi sum
• Gauss sum

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• PlanetMath page