Stickelberger's theorem

Let K_m 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 G_m isomorphic to the multiplicative group of integers modulo m (ℤ/mℤ)^×. The Stickelberger element (of level m or of K_m) is an element in the group ring ℚ[G_m] and the Stickelberger ideal (of level m or of K_m) is an ideal in the group ring ℤ[G_m]. They are defined as follows. Let ζ_m denote a primitive mth root of unity. The isomorphism from (ℤ/mℤ)^× to G_m is given by sending a to σ_a defined by the relation

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

The Stickelberger element of level 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 m, denoted I(K_m), is the set of integral multiples of θ(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 F be any Abelian number field whose Galois group over 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 ℚ). There is a natural group homomorphism G_m → G_F given by restriction, i.e. if σ ∈ G_m, its image in G_F is its restriction to F denoted res_mσ. The Stickelberger element of 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 F, denoted I(F), is defined as in the case of K_m, i.e.

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

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 ℤ/mℤ. This not true for general F.[2]

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[G_F] is.

Explicitly, the theorem is saying that if α ∈ 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 J is any fractional ideal of 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

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• PlanetMath page