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 OK mod Φ.
Lifts of the Frobenius are in correspondence with p-derivations.
Read more about this topic: Frobenius Endomorphism
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