Stickelberger's theorem explained

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 (Kummer|1847}}|1847) while the general result is due to Ludwig Stickelberger (Stickelberger|1890}}|1890).

The Stickelberger element and the Stickelberger ideal

Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to

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

More generally, if be any Abelian number field whose Galois group over is denoted, then the Stickelberger element of and the Stickelberger ideal of can be defined. By the Kronecker–Weber theorem there is an integer such that is contained in . Fix the least such (this is the (finite part of the) conductor of over). There is a natural group homomorphism given by restriction, i.e. if, its image in is its restriction to denoted . The Stickelberger element of is then defined as

\theta(F) = (1)/(m) \underset{(a,m) =1 }{\sum ^{m}a ⋅ res} _{m\sigma} ^{-1} ^{a} \in\Q[G _{F].} The Stickelberger ideal of, denoted, is defined as in the case of, i.e.I(F) = \theta(F)\Z[G _{F]\cap\Z[G} _{F].}

In the special case where, the Stickelberger ideal is generated by as varies over . This not true for general F.^[1]

Examples

If is a totally real field of conductor, then

\theta(F) = (\varphi(m))/(2 [F:\Q]) \sum _{\sigma\in G _{F}} \sigma, where is the Euler totient function and is the degree of over Q .

Statement of the theorem

Stickelberger's Theorem
Let be an abelian number field. Then, the Stickelberger ideal of annihilates the class group of .

Note that itself need not be an annihilator, but any multiple of it in is.

Explicitly, the theorem is saying that if is such that

\alpha\theta(F) = \sum _{\sigma\in G _{F}} a _{\sigma\sigma\in\Z[G} _{F]} and if is any fractional ideal of, then\prod _{\sigma\in G _{F}} \sigma\left(J ^{a _{\sigma}} \right) is a principal ideal.

See also

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

References

• Book: Cohen, Henri . Henri Cohen (number theorist)

. Henri Cohen (number theorist) . 2007 . Number Theory – Volume I: Tools and Diophantine Equations . 978-0-387-49922-2 . . . 239. 1119.11001 . 150–170 .

• Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
• Book: Fröhlich, A. . 0376.12002 . Albrecht Fröhlich . Stickelberger without Gauss sums . 589–607 . Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 . Fröhlich . A. . Albrecht Fröhlich . Academic Press . 1977 . 0-12-268960-7 .
• Book: Ireland . Kenneth . Rosen . Michael . A Classical Introduction to Modern Number Theory . New York . 2nd . Springer-Verlag . Graduate Texts in Mathematics . 84 . 1990 . 978-1-4419-3094-1 . 1070716 . 10.1007/978-1-4757-2103-4.

External links

• PlanetMath page

Notes and References

1. , Lemma 6.9 and the comments following it