Artin map
Let be a Galois extension of number fields, with rings of integers and . For any finite prime lying over a prime , let denote the decomposition group of , let denote the inertia group of , and let and be the residue fields. The exact sequence
yields an isomorphism . In particular, there is a unique element in , denoted , which maps to the power Frobenius map under this isomorphism (where is the number of elements in ). The notation is referred to as the Artin symbol of the extension at .
If we add the additional assumption that is unramified, then is the trivial group, and in this situation is an element of , called the Frobenius automorphism of .
If, furthermore, is an abelian extension (that is, is an abelian group), then for any other prime lying over . In this case, the Frobenius automorphism is denoted ; the change in notation from to reflects the fact that the automorphism is determined by independent of which prime of above it is chosen for use in the above construction.
Definition 1.
Let be a finite set of primes of , containing all the primes that ramify in . Let denote the subgroup of the group of fractional ideals of which is generated by all the primes in that are not in . The Artin map
is the map given by for all primes , extended linearly to .
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 |