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Stickelberger's theorem

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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 groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890).

Contents

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

σ a ( ζ m ) = ζ m a .

The Stickelberger element of level m is defined as

θ ( K m ) = 1 m ∑ a = 1 m ( a , m ) = 1 a ⋅ σ a − 1 ∈ Q [ G m ] .

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 ( K m ) = θ ( K m ) Z [ G m ] ∩ Z [ G m ] .

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 ) = 1 m ∑ a = 1 m ( a , m ) = 1 a ⋅ r e s m σ a − 1 ∈ Q [ G F ] .

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

I ( F ) = θ ( F ) Z [ G F ] ∩ Z [ G F ] .

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.

Examples

If F is a totally real field of conductor m, then

θ ( F ) = φ ( m ) 2 [ F : Q ] ∑ σ ∈ G F σ ,

where φ is the Euler totient function and [F : ℚ] is the degree of F over ℚ.

Statement of the theorem

Stickelberger's Theorem
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

α θ ( F ) = ∑ σ ∈ G F a σ σ ∈ Z [ G F ]

and if J is any fractional ideal of F, then

∏ σ ∈ G F σ ( J a σ )

is a principal ideal.

References

Stickelberger's theorem Wikipedia
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