# 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 $\U0001d513\subset L$ lying over a prime $\U0001d52d\in K$, let $D(\U0001d513)$ denote the decomposition group^{} of $\U0001d513$, let $T(\U0001d513)$ denote the inertia group of $\U0001d513$, and let $l:={\mathcal{O}}_{L}/\U0001d513$ and $k:={\mathcal{O}}_{K}/\U0001d52d$ be the residue fields^{}. The exact sequence^{}

$$1\u27f6T(\U0001d513)\u27f6D(\U0001d513)\u27f6\mathrm{Gal}(l/k)\u27f61$$ |

yields an isomorphism^{} $D(\U0001d513)/T(\U0001d513)\cong \mathrm{Gal}(l/k)$. In particular, there is a unique element in $D(\U0001d513)/T(\U0001d513)$, denoted $[L/K,\U0001d513]$, which maps to the ${q}^{\mathrm{th}}$ power Frobenius map^{} ${\mathrm{Frob}}_{q}\in \mathrm{Gal}(l/k)$ under this isomorphism (where $q$ is the number of elements in $k$). The notation $[L/K,\U0001d513]$ is referred to as the Artin symbol^{} of the extension^{} $L/K$ at $\U0001d513$.

If we add the additional assumption^{} that $\U0001d52d$ is unramified, then $T(\U0001d513)$ is the trivial group, and $[L/K,\U0001d513]$ in this situation is an element of $D(\U0001d513)\subset \mathrm{Gal}(L/K)$, called the Frobenius automorphism^{} of $\U0001d513$.

If, furthermore, $L/K$ is an abelian extension^{} (that is, $\mathrm{Gal}(L/K)$ is an abelian group^{}), then $[L/K,\U0001d513]=[L/K,{\U0001d513}^{\prime}]$ for any other prime ${\U0001d513}^{\prime}\subset L$ lying over $\U0001d52d$. In this case, the Frobenius automorphism $[L/K,\U0001d513]$ is denoted $(L/K,\U0001d52d)$; the change in notation from $\U0001d513$ to $\U0001d52d$ reflects the fact that the automorphism^{} is determined by $\U0001d52d\in K$ independent of which prime $\U0001d513$ 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

$${\varphi}_{L/K}:{I}_{K}^{S}\u27f6\mathrm{Gal}(L/K)$$ |

is the map given by ${\varphi}_{L/K}(\U0001d52d):=(L/K,\U0001d52d)$ for all primes $\U0001d52d\notin S$, 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 |