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, 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.
|Date of creation||2013-03-22 12:34:55|
|Last modified on||2013-03-22 12:34:55|
|Last modified by||djao (24)|