In
mathematics, Stickelberger's theorem is a result of
algebraic number theory, which gives some information about the
Galois module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
structure of
class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
s of
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
s. A special case was first proven by
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned ...
(
1847
Events
January–March
* January 4 – Samuel Colt sells his first revolver pistol to the U.S. government.
* January 13 – The Treaty of Cahuenga ends fighting in the Mexican–American War in California.
* January 16 – John C. Frémont ...
) while the general result is due to
Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomi ...
(
1890).
The Stickelberger element and the Stickelberger ideal
Let denote the th
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
, i.e. the
extension of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s obtained by
adjoining the th
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
to
(where is an integer). It is a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
of
with
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
isomorphic to the
multiplicative group of integers modulo . The Stickelberger element (of level or of ) is an element in the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
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
:
.
The Stickelberger element of level is defined as
:
The Stickelberger ideal of level , denoted , is the set of integral multiples of which have integral coefficients, i.e.
:
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
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...
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
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
given by restriction, i.e. if , its image in is its restriction to denoted . The Stickelberger element of is then defined as
:
The Stickelberger ideal of , denoted , is defined as in the case of , i.e.
:
In the special case where , the Stickelberger ideal is generated by as varies over . This not true for general ''F''.
Examples
If is a
totally real field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...
of conductor , then
:
where is the
Euler totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In o ...
and is the
degree of over
.
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
: