Frobenius Endomorphism - Frobenius For Global Fields

Frobenius For Global Fields

In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K. Since the extension is unramified the decomposition group of Φ is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of L as in the local case, by

where q is the order of the residue field O_K mod Φ.

Lifts of the Frobenius are in correspondence with p-derivations.