In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate. In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, class field theory is a major branch of algebraic number theory. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ...

Definitions

A formation is a topological group G together with a G-module A. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...

A layer E/F of a formation is a pair of open subgroups E, F such that F is a subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then A^E is defined to be the elements of A fixed by E. We write

Hⁿ(E/F)

for the Tate cohomology group Hⁿ(E/F, A^F) whenever E/F is a normal layer. In applications, G is usually the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ...

A class formation is a formation such that for every normal layer E/F

H¹(E/F) is trivial, and

H²(E/F) is cyclic of order |E/F|.

In practice, these cyclic groups come provided with canonical generators u_E/F ∈ H²(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation.

A formation that satisifies just the condition H¹(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert's theorem 90. In number theory, Hilberts Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if α is an element of L of relative norm 1, then then there exists β in L such that α = β/βs. ...

Examples of class formations

The most important examples of class formations (arranged roughly in order of difficulty) are as follows:

• Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.
• Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers.
• Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
• Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group.
• Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group.
• Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A.

It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below. In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only...

The first inequality

The first inequality of class field theory^[1] states that

H⁰(E/F) ≥ |E/F|

for cyclic layers E/F. It is usually proved using properties of the Herbrand quotient, in the more precise form

|H⁰(E/F)| = |E/F|×|H¹(E/F)|.

It is fairly straighforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules.

Before about 1950, the first inequality was known as the second inequality.

The second inequality

The second inequality of class field theory states that

H⁰(E/F) ≤ |E/F|

for all normal layers E/F.

For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology. In number theory, Hilberts Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if α is an element of L of relative norm 1, then then there exists β in L such that α = β/βs. ...

The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler's proof that there are infinitely many primes using the pole at s=1 of the Riemann zeta function.) As each prime in k that is a norm is the product of deg(K/k)= |E/F| distinct degree 1 primes of K, this shows that the set of primes of k that are norms has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H⁰(E/F), one shows that the Dirichlet density of primes of k representing the trivial element of this group has density 1/|H⁰(E/F)|. (This part of the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group H⁰(E/F) if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So In mathematics, the Dedekind zeta-function is a Dirichlet series defined for any algebraic number field , and denoted where is a complex variable. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

1/|H⁰(E/F)| ≥ 1/|E/F|

which is the second inequality.

In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality.

Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, H⁰(E/F) is isomorphic to the abelianization of E/F, so equality in the second inequality holds exactly for abelain extensions, and class fields are the same as abelian extensions.

The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that

H¹(E/F)|E/F| = H⁰(E/F) ≤ |E/F|

so

H⁰(E/F) = |E/F|

and

H¹(E/F) = 1.

Now a basic theorem about cohomology groups shows that since H¹(E/F) = 1 for all cyclic layers, we have

H¹(E/F) = 1

for all normal layers (so in particular the formation is a field formation). This proof that H¹(E/F) is always trivial is rather roundabout; no "direct" proof of it (whatever this means) for global fields is known. (For local fields the vanishing of H¹(E/F) is just Hilbert's theorem 90.) In number theory, Hilberts Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if α is an element of L of relative norm 1, then then there exists β in L such that α = β/βs. ...

For cyclic group, H⁰ is the same as H², so H²(E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H¹(E/F) = 1 for all normal layers and H²(E/F) ≤ |E/F| for all cyclic layers, we have

H²(E/F)≤ |E/F|

for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)

The Brauer group

The Brauer groups H²(E/*) of a class formation are defined to be the direct limit of the groups H²(E/F) as F runs over all open subgroups of E. An easy consequence of the vanishing of H¹ for all layers is that the groups H²(E/F) are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related). In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ...

The next step is to prove that H²(E/F) is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H²(E/F).

For cyclic extensions this is already known. The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n. These homomorphisms are constructed using cyclic cyclotomic extensions. This idea was first used by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem. In mathematics, Chebotarevs density theorem in algebraic number theory is a generalisation to algebraic number fields that are Galois extensions, of Dirichlets theorem on arithmetic progressions. ...

The proof of the existence of an element of order |E/F| for an arbitrary layer proceeds by first constructing a suitable auxiliary cyclic extension of degree |E/F| as above; as this is cyclic, there is an element of order |E/F| in its second cohomology, and this element turns out to be essentially an element of H²(E/F).

This shows that the second cohomology group H²(E/F) of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation. With a little more care in the proofs, we get a canonical generator of H²(E/F), called the fundamental class.

It follows from this that the Brauer group H²(E/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.

Tate's theorem and the Artin map

Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H²(G,A), such that for every subgroup E of G

• H¹(E,A) is trivial, and
• H²(E,A) is generated by Res(a) which has order E.

Then cup product with a is an isomorphism

• Hⁿ(G,Z) → Hⁿ⁺²(G,A).

If we apply the case n=−2 of Tate's theorem to a class formation, we find that there is an isomorphism

• H⁻²(E/F,Z) → H⁰(E/F,A^F)

for any normal layer E/F. The group H⁻²(E/F,Z) is just the abelianization of E/F, and the group H⁰(E/F,A^F) is A^E modulo the group of norms of A^F. In other words we have an explicit description of the abelianization of the Galois group E/F in terms of A^E.

Taking the inverse of this isomorphism gives a homomorphism

A^E → abelianization of E/F,

and taking the limit over all open subgroups F gives a homomophism

A^E → abelianization of E,

called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of A^E (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.

The Takagi existence theorem

The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by first adding in lots of roots of unity, and then taking Kummer extensions. These extensions may be non-abelian (though they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group. In class field theory, the Takagi existence theorem states in part that if K is a number field with class group G, there exists a unique abelian extension L/K with Galois group G, such that every ideal in K becomes principal in L, and that L is characterized by... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots...

A consequence is that the group H⁰(F, A^F) is exactly the idele class group modulo the connected component of the identity, or equivalently the profinite completion of the idele class group. By the Artin isomorphism, this is the abelianization of the Galois group of F.

In the case of characteristic p>0, we need to use Artin-Schreier extensions as well as Kummer extensions.

For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin-Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in many cases, but a general method for constructing all abelian extensions directly (without first constructing a larger metabelian extension) is not known.

Weil group

This is not the Weyl group and has no connection with the Weil-Châtelet group or the Mordell-Weil group

The Weil group of a class formation with fundamental classes u_E/F ∈ H²(E/F, A^F) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands programme. In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ...

If E/F is a normal layer, then the Weil group U of E/F is the extension

1 → A^F → U → E/F → 1

corresponding to the fundamental class u_E/F in H²(E/F, A^F). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G.

For archimedean local fields the Weil group is easy to describe: for C it is the group C^* of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C^* ∪ j C^* of the non-zero quaternions.

For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse). In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ...

For local or global fields of characteristic p>0, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).

For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, consisting of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.

For number fields there is no known "natural" construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.

A more refined, later version of the construction is the Weil-Deligne group scheme, an extension of the Weil group by a 1 dimensional additive group scheme.

For more details about Weil groups see the Artin-Tate notes, or

• Tate, J. Number theoretic background. Automorphic forms, representations, and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 3-26, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, ISBN 0-8218-1435-4
• Weil, A Sur la theorie du corps de classes (On class field theory), J. Math. Soc. Japan 3 (1951) 1-35, reprinted in volume I of his collected papers, ISBN 0-387-90330-5

See also

• Abelian extension
• Artin L-function
• Artin reciprocity
• Class field theory
• Complex multiplication
• Galois cohomology
• Hasse norm theorem
• Herbrand quotient
• Hilbert class field
• Kronecker–Weber theorem
• Local classfield theory
• Takagi existence theorem
• Tate cohomology group

In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1920s by Emil Artin, in connection with his research into class field theory. ... In mathematics, Artin reciprocity refers to various results connecting Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to Heckes grossencharacters of that number field. ... In mathematics, class field theory is a major branch of algebraic number theory. ... In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at... In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ... In number theory, the Hasse norm theorem tells us that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. ... In algebraic number theory, the Hilbert class field E of a number field is the maximal abelian unramified extension of Note that in this context, unramified is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. ... In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. ... In class field theory, the Takagi existence theorem states in part that if K is a number field with class group G, there exists a unique abelian extension L/K with Galois group G, such that every ideal in K becomes principal in L, and that L is characterized by...

Notes

1. ^ For the nomenclature, note that 'first' and 'second' switched over at some point c.1950. What is now the 'second' was once the 'first' (see for example p. 49 in this treatment (PDF); this bounds the index of the norms in a class group, in old-fashioned language, and is the part of the main proof that was initially treated by means of L-functions. The historical reason behind this is that the first inequality of genus theory (concerned with 2-torsion in the class groups of quadratic fields) was an upper bound for the number of genera. (discussed at introduction to the Hilbert Zahlbericht (PDF).

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ...

References

• E. Artin, J. Tate, Class field theory ISBN 0-201-51011-1