Artin reciprocity

The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" 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 to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.

Statement

Let

L

/

K

{\displaystyle L/K}

be a Galois extension of global fields,

Gal

(

L

/

K

)

{\displaystyle {\text{Gal}}(L/K)}

be Galois group of this extension and

C

K

,

C

L

{\displaystyle C_{K},C_{L}}

denote the idèle class groups

of respectively

K

{\displaystyle K}

and

L

{\displaystyle L}

. Let:

N

L

/

K

:

C

L

→

C

K

{\displaystyle N_{L/K}:C_{L}\rightarrow C_{K}}

denote idele norm map. The Artin reciprocity law states that there exist canonical isomorphism between quotient group and abelianization of Galois group:

θ

:

C

K

/

N

L

/

K

(

C

L

)

→

Gal

⁡

(

L

/

K

)

ab

,

{\displaystyle \theta :C_{K}/{N_{L/K}(C_{L})}\to \operatorname {Gal} (L/K)^{\text{ab}},}

that is defined by assembling the local maps:

θ

v

:

K

v

×

/

N

L

v

/

K

v

(

L

v

×

)

→

Gal

(

L

/

K

)

ab

,

{\displaystyle \theta _{v}:K_{v}^{\times }/N_{L_{v}/K_{v}}(L_{v}^{\times })\to {\text{Gal}}(L/K)^{\text{ab}},}

for different places

v

{\displaystyle v}

of

K

{\displaystyle K}

.

Remarks

The map

θ

{\displaystyle \theta }

is called global symbol map or Artin symbol.

For each place

v

{\displaystyle v}

the local map

θ

v

{\displaystyle \theta _{v}}

is called local Artin symbol, local reciprocity map or norm residue symbol.

The local maps

θ

v

{\displaystyle \theta _{v}}

are isomorphisms themselves, this is the content of local reciprocity law, key result in local class field theory.

Proof

A cohomological proof of the global reciprocity law can be achieved by first establishing that

(

Gal

⁡

(

K

sep

/

K

)

,

lim

→

⁡

C

L

)

{\displaystyle (\operatorname {Gal} (K^{\text{sep}}/K),\varinjlim C_{L})}

constitutes a class formation in the sense of Artin and Tate.