The Artin reciprocity law, which was established by
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
in a series of papers (1924; 1927; 1930), is a general theorem in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
that forms a central part of global
class field theory. The term "
reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...
" refers to a long line of more concrete number theoretic statements which it generalized, from the
quadratic reciprocity law and the reciprocity laws of
Eisenstein and
Kummer Kummer is a German surname. Notable people with the surname include:
*Bernhard Kummer (1897–1962), German Germanist
*Clare Kummer (1873—1958), American composer, lyricist and playwright
*Clarence Kummer (1899–1930), American jockey
* Christo ...
to
Hilbert's product formula for the
norm symbol. Artin's result provided a partial solution to
Hilbert's ninth problem.
Statement
Let
be 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 fiel ...
of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s and
stand for the
idèle class group
of
. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map
[Neukirch (1999) p.391]
:
where
denotes the
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of a group. The map
is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the
norm residue symbol[Serre (1967) p.140][Serre (1979) p.197]
:
for different places
of
. More precisely,
is given by the local maps
on the
-component of an idèle class. The maps
are isomorphisms. This is the content of the ''local reciprocity law'', a main theorem of
local class field theory.
Proof
A cohomological proof of the global reciprocity law can be achieved by first establishing that
:
constitutes a
class formation in the sense of Artin and Tate.
[Serre (1979) p.164] Then one proves that
:
where
denote the
Tate cohomology groups. Working out the cohomology groups establishes that
is an isomorphism.
Significance
Artin's reciprocity law implies a description of the
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of the absolute
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 ...
of a
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function fi ...
''K'' which is based on the
Hasse local–global principle and the use of the
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
s. Together with the
Takagi existence theorem, it is used to describe the
abelian extensions of ''K'' in terms of the arithmetic of ''K'' and to understand the behavior of the
nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that
Artin L-functions are
meromorphic and for the proof of the
Chebotarev density theorem.
Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the
transfer homomorphism of I. Schur and used the reciprocity law to translate the
principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.
Finite extensions of global fields
The definition of the Artin map for a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
abelian extension ''L''/''K'' of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s (such as a finite abelian extension of
) has a concrete description in terms of
prime ideals and
Frobenius element
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
s.
If
is a prime of ''K'' then the
decomposition group In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O'K'' factorise as products of prime ideals of ''O'L'', provides one ...
s of primes
above
are equal in Gal(''L''/''K'') since the latter group is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
. If
is
unramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) ...
in ''L'', then the decomposition group
is canonically isomorphic to the Galois group of the extension of residue fields
over
. There is therefore a canonically defined Frobenius element in Gal(''L''/''K'') denoted by
or
. If Δ denotes the
relative discriminant
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume ...
of ''L''/''K'', the Artin symbol (or Artin map, or (global) reciprocity map) of ''L''/''K'' is defined on the
group of prime-to-Δ fractional ideals,
, by linearity:
:
The Artin reciprocity law (or global reciprocity law) states that there is a
modulus c of ''K'' such that the Artin map induces an isomorphism
:
where ''K''
c,1 is the
ray modulo c, N
''L''/''K'' is the norm map associated to ''L''/''K'' and
is the fractional ideals of ''L'' prime to c. Such a modulus c is called a defining modulus for ''L''/''K''. The smallest defining modulus is called the
conductor of ''L''/''K'' and typically denoted
Examples
Quadratic fields
If
is a
squarefree integer,
and
, then
can be identified with . The discriminant Δ of ''L'' over
is ''d'' or 4''d'' depending on whether ''d'' ≡ 1 (mod 4) or not. The Artin map is then defined on primes ''p'' that do not divide Δ by
:
where
is the
Kronecker symbol.
More specifically, the conductor of
is the principal ideal (Δ) or (Δ)∞ according to whether Δ is positive or negative, and the Artin map on a prime-to-Δ ideal (''n'') is given by the Kronecker symbol
This shows that a prime ''p'' is split or inert in ''L'' according to whether
is 1 or −1.
Cyclotomic fields
Let ''m'' > 1 be either an odd integer or a multiple of 4, let
be a
primitive ''m''th root of unity, and let
be the ''m''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 o ...
.
can be identified with
by sending σ to ''a''
σ given by the rule
:
The conductor of
is (''m'')∞, and the Artin map on a prime-to-''m'' ideal (''n'') is simply ''n'' (mod ''m'') in
Relation to quadratic reciprocity
Let ''p'' and
be distinct odd primes. For convenience, let
(which is always 1 (mod 4)). Then, quadratic reciprocity states that
:
The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field
and the cyclotomic field
as follows.
First, ''F'' is a subfield of ''L'', so if ''H'' = Gal(''L''/''F'') and
then
Since the latter has order 2, the subgroup ''H'' must be the group of squares in
A basic property of the Artin symbol says that for every prime-to-ℓ ideal (''n'')
:
When ''n'' = ''p'', this shows that
if and only if, ''p'' modulo ℓ is in ''H'', i.e. if and only if, ''p'' is a square modulo ℓ.
Statement in terms of ''L''-functions
An alternative version of the reciprocity law, leading to the
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic n ...
, connects
Artin L-functions associated to abelian extensions of a
number field with Hecke L-functions associated to characters of the idèle class group.
[James Milne]
''Class Field Theory''
/ref>
A Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ...
(or Größencharakter) of a number field ''K'' is defined to be a quasicharacter In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualif ...
of the idèle class group of ''K''. Robert Langlands
Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study ...
interpreted Hecke characters as automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s on the reductive algebraic group ''GL''(1) over the ring of adeles of ''K''.[.]
Let be an abelian Galois extension 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 ...
''G''. Then for any character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
(i.e. one-dimensional complex representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of the group ''G''), there exists a Hecke character of ''K'' such that
:
where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated with χ, Section 7.D of.[
The formulation of the Artin reciprocity law as an equality of ''L''-functions allows formulation of a generalisation to ''n''-dimensional representations, though a direct correspondence is still lacking.
]
Notes
References
*Emil Artin (1924) "Über eine neue Art von L-Reihen", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 3: 89–108; ''Collected Papers'', Addison Wesley
Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throug ...
(1965), 105–124
*Emil Artin (1927) "Beweis des allgemeinen Reziprozitätsgesetzes", ''Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg'' 5: 353–363; ''Collected Papers'', 131–141
*Emil Artin (1930) "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes", ''Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg'' 7: 46–51; ''Collected Papers'', 159–164
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*{{citation , last=Tate , first=John , authorlink=John Tate (mathematician) , chapter=VII. Global class field theory , pages=162–203 , editor1-last=Cassels , editor1-first=J.W.S. , editor1-link=J. W. S. Cassels , editor2-last=Fröhlich , editor2-first=A. , editor2-link=Albrecht Fröhlich , title=Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union , location=London , publisher=Academic Press , year=1967 , zbl=0153.07403
Class field theory