the action of Frobenius on x is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism is the same as the composite morphism: by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of X. Frobenius for local fields Given an unramified finite extension of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields. Suppose is an unramified extension of local fields, with ring of integers OK of such that the residue field, the integers of modulo their unique maximal ideal , is a finite field of order , where is a power of a prime. If is a prime of lying over , that is unramified means by definition that the integers of modulo , the residue field of , will be a finite field of order extending the residue field of where is the degree of . We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that Frobenius for global fields In algebraic number theory, Frobenius elements are defined for extensions of global fields that are finite Galois extensions for prime ideals of that are unramified in . Since the extension is unramified the decomposition group of 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 as in the local case, by where is the order of the residue field . Lifts of the Frobenius are in correspondence with p-derivations. Examples The polynomial has discriminant , and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root of it to the field of -adic numbers gives an unramified extension of . We may find the image of under the Frobenius map by locating the root nearest to , which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the -adic integers . Modulo this polynomial is . This is algebraic over and is the correct global Frobenius image in terms of the embedding of into ;