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Artin map (Definition)

Let $ L/K$ be a Galois extension of number fields, with rings of integers $ \O _L$ and $ \O _K$. For any finite prime $ \P\subset L$ lying over a prime $ \mathfrak{p}\in K$, let $ D(\P )$ denote the decomposition group of $ \P$, let $ T(\P )$ denote the inertia group of $ \P$, and let $ l := \O _L/\P$ and $ k := \O _K/\mathfrak{p}$ be the residue fields. The exact sequence

$\displaystyle 1 \longrightarrow T(\P ) \longrightarrow D(\P ) \longrightarrow \operatorname{Gal}(l/k) \longrightarrow 1 $
yields an isomorphism $ D(\P )/T(\P ) \cong \operatorname{Gal}(l/k)$. In particular, there is a unique element in $ D(\P )/T(\P )$, denoted $ [L/K,\P ]$, which maps to the $ q^{\rm th}$ power Frobenius map $ \operatorname{Frob}_q \in \operatorname{Gal}(l/k)$ under this isomorphism (where $ q$ is the number of elements in $ k$). The notation $ [L/K,\P ]$ is referred to as the Artin symbol of the extension $ L/K$ at $ \P$.

If we add the additional assumption that $ \mathfrak{p}$ is unramified, then $ T(\P )$ is the trivial group, and $ [L/K,\P ]$ in this situation is an element of $ D(\P ) \subset \operatorname{Gal}(L/K)$, called the Frobenius automorphism of $ \P$.

If, furthermore, $ L/K$ is an abelian extension (that is, $ \operatorname{Gal}(L/K)$ is an abelian group), then $ [L/K,\P ] = [L/K,\P ']$ for any other prime $ \P ' \subset L$ lying over $ \mathfrak{p}$. In this case, the Frobenius automorphism $ [L/K,\P ]$ is denoted $ (L/K,\mathfrak{p})$; the change in notation from $ \P$ to $ \mathfrak{p}$ reflects the fact that the automorphism is determined by $ \mathfrak{p}\in K$ independent of which prime $ \P$ of $ L$ above it is chosen for use in the above construction.

Definition 1   Let $ S$ be a finite set of primes of $ K$, containing all the primes that ramify in $ L$. Let $ I_K^S$ denote the subgroup of the group $ I_K$ of fractional ideals of $ K$ which is generated by all the primes in $ K$ that are not in $ S$. The Artin map
$\displaystyle \phi_{L/K}: I_K^S \longrightarrow \operatorname{Gal}(L/K) $
is the map given by $ \phi_{L/K}(\mathfrak{p}) := (L/K,\mathfrak{p})$ for all primes $ \mathfrak{p}\notin S$, extended linearly to $ I_K^S$.



"Artin map" is owned by djao.
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See Also: ray class field

Also defines:  Artin symbol, Frobenius automorphism
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Cross-references: generated by, fractional ideals, subgroup, finite set, independent, automorphism, reflects, abelian group, abelian extension, group, unramified, extension, number, Frobenius map, maps, isomorphism, exact sequence, residue fields, inertia group, decomposition group, prime, finite prime, rings of integers, number fields, Galois extension
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This is version 6 of Artin map, born on 2002-04-14, modified 2005-03-15.
Object id is 2831, canonical name is ArtinMap.
Accessed 6607 times total.

Classification:
AMS MSC: 11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)
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