(See math.stackexchange.com for an explanation of some of the terms used here) The definition of the Artin map for a
finite abelian extension L/
K of
global fields (such as a finite abelian extension of \Q) has a concrete description in terms of
prime ideals and
Frobenius elements. If \mathfrak{p} is a prime of
K then the
decomposition groups of primes \mathfrak{P} above \mathfrak{p} are equal in Gal(
L/
K) since the latter group is
abelian. If \mathfrak{p} is
unramified in
L, then the decomposition group D_\mathfrak{p} is canonically isomorphic to the Galois group of the extension of residue fields \mathcal{O}_{L,\mathfrak{P}}/\mathfrak{P} over \mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}. There is therefore a canonically defined Frobenius element in Gal(
L/
K) denoted by \mathrm{Frob}_\mathfrak{p} or \left(\frac{L/K}{\mathfrak{p}}\right). If Δ denotes the
relative discriminant 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, I_K^\Delta, by linearity: :\begin{cases} \left(\frac{L/K}{\cdot}\right):I_K^\Delta \longrightarrow \operatorname{Gal}(L/K)\\ \prod_{i=1}^m\mathfrak{p}_i^{n_i} \longmapsto \prod_{i=1}^m\left(\frac{L/K}{\mathfrak{p}_i}\right)^{n_i} \end{cases} 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 :I_K^\mathbf{c}/i(K_{\mathbf{c},1})\mathrm{N}_{L/K}(I_L^\mathbf{c})\overset{\sim}{\longrightarrow}\mathrm{Gal}(L/K) where
Kc,1 is the
ray modulo c, N
L/
K is the norm map associated to
L/
K and I_L^\mathbf{c} 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 \mathfrak{f}(L/K).
Examples Quadratic fields If d\neq1 is a
squarefree integer, K=\Q, and L=\Q(\sqrt{d}), then \operatorname{Gal}(L/\Q) can be identified with {±1}. The discriminant Δ of
L over \Q 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 :p\mapsto\left(\frac{\Delta}{p}\right) where \left(\frac{\Delta}{p}\right) is the
Kronecker symbol. More specifically, the conductor of L/\Q 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 \left(\frac{\Delta}{n}\right). This shows that a prime
p is split or inert in
L according to whether \left(\frac{\Delta}{p}\right) is 1 or −1.
Cyclotomic fields Let
m > 1 be either an odd integer or a multiple of 4, let \zeta_m be a
primitive mth root of unity, and let L = \Q(\zeta_m) be the
mth
cyclotomic field. \operatorname{Gal}(L/\Q) can be identified with (\Z/m\Z)^{\times} by sending σ to
aσ given by the rule :\sigma(\zeta_m)=\zeta_m^{a_\sigma}. The conductor of L/\Q is (
m)∞, and the Artin map on a prime-to-
m ideal (
n) is simply
n (mod
m) in (\Z/m\Z)^{\times}.
Relation to quadratic reciprocity Let
p and \ell be distinct odd primes. For convenience, let \ell^* = (-1)^{\frac{\ell-1}{2}}\ell (which is always 1 (mod 4)). Then, quadratic reciprocity states that :\left(\frac{\ell^*}{p}\right)=\left(\frac{p}{\ell}\right). The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field F=\Q(\sqrt{\ell^*}) and the cyclotomic field L=\Q(\zeta_\ell) as follows. First,
F is a subfield of
L, so if
H = Gal(
L/
F) and G= \operatorname{Gal}(L/\Q), then \operatorname{Gal}(F/\Q) = G/H. Since the latter has order 2, the subgroup
H must be the group of squares in (\Z/\ell\Z)^{\times}. A basic property of the Artin symbol says that for every prime-to-ℓ ideal (
n) :\left(\frac{F/\Q}{(n)}\right)=\left(\frac{L/\Q}{(n)}\right)\pmod H. When
n =
p, this shows that \left(\frac{\ell^*}{p}\right)=1 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 ==