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 $\\mathfrak{p}\otin S$ , extended linearly to $I_{K}^{S}$ .