# 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 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 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 ArtinMap 2013-03-22 12:34:55 2013-03-22 12:34:55 djao (24) djao (24) 9 djao (24) Definition msc 11R37 RayClassField Artin symbol Frobenius automorphism