Stickelberger's theorem
Encyclopedia
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 field
s. A special case was first proven by Ernst Kummer
(Kummer) while the general result is due to Ludwig Stickelberger
(Stickelberger).
, i.e. the extension
of the rational number
s 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
The Stickelberger element of level m is defined as
The Stickelberger ideal of level m, denoted I(Km), is the set of integral multiples of θ(Km) which have integral coefficients, i.e.
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
The Stickelberger ideal of F, denoted I(F), is defined as in the case of Km, i.e.
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.
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
and if J is any fractional ideal
of F, then
is a principal ideal
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Stickelberger's theorem is a result of algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, which gives some information about the Galois module
Galois module
In mathematics, a Galois module is a G-module where G is 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, but can also be used as a synonym for G-module...
structure of class groups of cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s. A special case was first proven by Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
(Kummer) while the general result is due to Ludwig Stickelberger
Ludwig Stickelberger
Ludwig Stickelberger was a Swiss mathematician who made important contributions to linear algebra and algebraic number theory .-Short biography:Stickelberger was born in Buch in the canton of Schaffhausen into a family of a...
(Stickelberger).
The Stickelberger element and the Stickelberger ideal
Let Km denote the mth cyclotomic fieldCyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
, i.e. the extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s obtained by adjoining
Adjunction (field theory)
In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.- Definition :...
the mth roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
to Q (where m ≥ 2 is an integer). It is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
of Q with Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
Gm isomorphic to the multiplicative group of integers modulo m
Multiplicative group of integers modulo n
In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it...
(Z/mZ)×. The Stickelberger element (of level m or of Km) is an element in the group ring
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 one-to-one with the given group. As a ring, its addition law is that of the free...
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
- σa(ζ) = .
The Stickelberger element of level m is defined as
The Stickelberger ideal of level m, denoted I(Km), is the set of integral multiples of θ(Km) which have integral coefficients, i.e.
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
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
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
Conductor (class field theory)
In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.-Local conductor:...
of F over Q). There is a natural group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
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
The Stickelberger ideal of F, denoted I(F), is defined as in the case of Km, i.e.
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
-
- where φ is the Euler totient function and [F : Q] is the degreeDegree of a field extensionIn mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.-...
of F over Q.
Statement of the theorem
- Stickelberger's Theorem
- Let F be an abelian number field. Then, the Stickelberger ideal of F annihilatesAnnihilator (ring theory)In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
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
and if J is any fractional ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
of F, then
is a principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
.