Frobenius Endomorphism - Frobenius For Local Fields

Frobenius For Local Fields

The definition of F for schemes automatically defines F for local and global fields, but we will treat these cases separately for clarity.

The definition of the Frobenius for finite fields can be extended to other sorts of field extensions. Given an unramified finite extension L/K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.

Suppose L/K is an unramified extension of local fields, with ring of integers O_K of K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of order q. If Φ is a prime of L lying over φ, that L/K is unramified means by definition that the integers of L modulo Φ, the residue field of L, will be a finite field of order qf extending the residue field of K where f is the degree of L/K. We may define the Frobenius map for elements of the ring of integers O_L of L as an automorphism of L such that