Artin map

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

1\longrightarrow T(\mathfrak{P})\longrightarrow D(\mathfrak{P})\longrightarrow% \operatorname{Gal}(l/k)\longrightarrow 1

yields an isomorphism $D(\mathfrak{P})/T(\mathfrak{P})\cong\operatorname{Gal}(l/k)$ . In particular, there is a unique element in $D(\mathfrak{P})/T(\mathfrak{P})$ , denoted $[L/K,\mathfrak{P}]$ , 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 $[L/K,\mathfrak{P}]$ is referred to as the Artin symbol of the extension $L/K$ at $\mathfrak{P}$ .

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

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

\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 $I_{K}^{S}$ .

Title	Artin map
Canonical name	ArtinMap
Date of creation	2013-03-22 12:34:55
Last modified on	2013-03-22 12:34:55
Owner	djao (24)
Last modified by	djao (24)
Numerical id	9
Author	djao (24)
Entry type	Definition
Classification	msc 11R37
Related topic	RayClassField
Defines	Artin symbol
Defines	Frobenius automorphism