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).

The Stickelberger element and the Stickelberger ideal

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

<math display=block>\sigma_a(\zeta_m) = \zeta_m^a.</math>

The Stickelberger element of level <math>m</math> is defined as

<math display=block>\theta(K_m)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\sigma_a^{-1}\in\Q[G_m].</math>

The Stickelberger ideal of level <math>m</math>, denoted <math>I(K_m)</math>, is the set of integral multiples of <math>\theta(K_m)</math> which have integral coefficients, i.e.

<math display=block>I(K_m)=\theta(K_m)\Z[G_m]\cap\Z[G_m].</math>

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

<math display=block>\theta(F)=\frac{1}{m}\underset{(a,m)=1}{\sum_{a=1}^m}a\cdot\mathrm{res}_m\sigma_a^{-1}\in\Q[G_F].</math>

The Stickelberger ideal of <math>F</math>, denoted <math>I(F)</math>, is defined as in the case of <math>K_m</math>, i.e.

<math display=block>I(F)=\theta(F)\Z[G_F]\cap\Z[G_F].</math>

In the special case where <math>F=K_m</math>, the Stickelberger ideal <math>I(K_m)</math> is generated by <math>(a-\sigma_a)\theta(K_m)</math> as <math>a</math> varies over <math>\Z/m\Z</math>. This is not true for general <math>F</math>.

Examples

If <math>F</math> is a totally real field of conductor <math>m</math>, then

<math display=block>\theta(F)=\frac{\varphi(m)}{2[F:\Q]}\sum_{\sigma\in G_F}\sigma,</math>

where <math>\varphi</math> is the Euler totient function and <math>[F:\Q]</math> is the degree of <math>F</math> over <math>\Q</math>.

Statement of the theorem

<blockquote>Stickelberger's Theorem<br>

Let <math>F</math> be an abelian number field. Then, the Stickelberger ideal of <math>F</math> annihilates the class group of <math>F</math>.</blockquote>

Note that <math>\theta(F)</math> itself need not be an annihilator, but any multiple of it in <math>\Z[G_F]</math> is.

Explicitly, the theorem is saying that if <math>\alpha\in\Z[G_F]</math> is such that

<math display=block>\alpha\theta(F)=\sum_{\sigma\in G_F}a_\sigma\sigma\in\Z[G_F]</math>

and if <math>J</math> is any fractional ideal of <math>F</math>, then

<math display=block>\prod_{\sigma\in G_F}\sigma\left(J^{a_\sigma}\right)</math>

is a principal ideal.

See also

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

Notes

References

  • Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
  • PlanetMath page