Stickelberger's theorem

Let {{mvar|K_m}} denote the {{mvar|m}}th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the {{mvar|m}}th roots of unity to $\backslash mathbb\{Q\}$ (where {{math|m ≥ 2}} is an integer). It is a Galois extension of $\backslash mathbb\{Q\}$ with Galois group {{mvar|G_m}} isomorphic to the Multiplicative group of integers modulo n {{math|($\backslash mathbb\{Z\}$/m$\backslash mathbb\{Z\}$)^×}}. The Stickelberger element (of level {{mvar|m}} or of {{mvar|K_m}}) is an element in the group ring {{math|$\backslash mathbb\{Q\}$[G_m]}} and the Stickelberger ideal (of level {{mvar|m}} or of {{mvar|K_m}}) is an ideal in the group ring {{math|$\backslash mathbb\{Z\}$[G_m]}}. They are defined as follows. Let {{mvar|ζ_m}} denote a primitive root of unity. The isomorphism from {{math|($\backslash mathbb\{Z\}$/m$\backslash mathbb\{Z\}$)^×}} to {{mvar|G_m}} is given by sending {{mvar|a}} to {{mvar|σ_a}} defined by the relation

:$\backslash sigma\_a(\backslash zeta\_m)\; =\; \backslash zeta\_m^a$.

The Stickelberger element of level {{mvar|m}} is defined as

:$\backslash theta(K\_m)=\backslash frac\{1\}\{m\}\backslash underset\{(a,m)=1\}\{\backslash sum\_\{a=1\}^m\}a\backslash cdot\backslash sigma\_a^\{-1\}\backslash in\backslash Q[G\_m].$

The Stickelberger ideal of level {{mvar|m}}, denoted {{math|I(K_m)}}, is the set of integral multiples of {{math|θ(K_m)}} which have integral coefficients, i.e.

:$I(K\_m)=\backslash theta(K\_m)\backslash Z[G\_m]\backslash cap\backslash Z[G\_m].$

More generally, if {{mvar|F}} be any Abelian number field whose Galois group over {{math|$\backslash mathbb\{Q\}$}} is denoted {{mvar|G_F}}, then the Stickelberger element of {{mvar|F}} and the Stickelberger ideal of {{mvar|F}} can be defined. By the Kronecker–Weber theorem there is an integer {{mvar|m}} such that {{mvar|F}} is contained in {{mvar|K_m}}. Fix the least such {{mvar|m}} (this is the (finite part of the) conductor of {{mvar|F}} over {{math|$\backslash mathbb\{Q\}$}}). There is a natural group homomorphism {{math|G_m → G_F}} given by restriction, i.e. if {{math|σ ∈ G_m}}, its image in {{mvar|G_F}} is its restriction to {{mvar|F}} denoted {{math|res_mσ}}. The Stickelberger element of {{mvar|F}} is then defined as

:$\backslash theta(F)=\backslash frac\{1\}\{m\}\backslash underset\{(a,m)=1\}\{\backslash sum\_\{a=1\}^m\}a\backslash cdot\backslash mathrm\{res\}\_m\backslash sigma\_a^\{-1\}\backslash in\backslash Q[G\_F].$

The Stickelberger ideal of {{mvar|F}}, denoted {{math|I(F)}}, is defined as in the case of {{mvar|K_m}}, i.e.

:$I(F)=\backslash theta(F)\backslash Z[G\_F]\backslash cap\backslash Z[G\_F].$

In the special case where {{math|F {{=}} K_m}}, the Stickelberger ideal {{math|I(K_m)}} is generated by {{math|(a − σ_a)θ(K_m)}} as {{mvar|a}} varies over {{math|$\backslash mathbb\{Z\}$/m$\backslash mathbb\{Z\}$}}. This not true for general F.{{harvnb|Washington|1997}}, Lemma 6.9 and the comments following it

If {{mvar|F}} is a totally real field of conductor {{mvar|m}}, then{{harvnb|Washington|1997|loc=§6.2}}

:$\backslash theta(F)=\backslash frac\{\backslash varphi(m)\}\{2[F:\backslash Q]\}\backslash sum\_\{\backslash sigma\backslash in\; G\_F\}\backslash sigma,$

where {{mvar|φ}} is the Euler totient function and {{math|[F : $\backslash mathbb\{Q\}$]}} is the degree of {{mvar|F}} over $\backslash mathbb\{Q\}$.

Stickelberger's Theorem{{harvnb|Washington|1997|loc=Theorem 6.10}}

Let {{mvar|F}} be an abelian number field. Then, the Stickelberger ideal of {{mvar|F}} annihilates the class group of {{mvar|F}}.

Note that {{math|θ(F)}} itself need not be an annihilator, but any multiple of it in {{math|$\backslash mathbb\{Z\}$[G_F]}} is.

Explicitly, the theorem is saying that if {{math|α ∈ $\backslash mathbb\{Z\}$[G_F]}} is such that

:$\backslash alpha\backslash theta(F)=\backslash sum\_\{\backslash sigma\backslash in\; G\_F\}a\_\backslash sigma\backslash sigma\backslash in\backslash Z[G\_F]$

and if {{mvar|J}} is any fractional ideal of {{mvar|F}}, then

:$\backslash prod\_\{\backslash sigma\backslash in\; G\_F\}\backslash sigma\backslash left(J^\{a\_\backslash sigma\}\backslash right)$

is a principal ideal.

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

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Category:Cyclotomic fields

Category:Theorems in algebraic number theory