Arithmetic and Geometric Frobenius
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High Quality Content by WIKIPEDIA articles! In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping that takes r in R to rp is a ring endomorphism of R. The image of is then Rp, the subring of R consisting of p-th powers. In some important cases, for example finite fields, is surjective. Otherwise is an endomorphism but not a ring automorphism. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to . This gives a mapping : Spec(Rp) Spec(R) of affine scheme...
High Quality Content by WIKIPEDIA articles! In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping that takes r in R to rp is a ring endomorphism of R. The image of is then Rp, the subring of R consisting of p-th powers. In some important cases, for example finite fields, is surjective. Otherwise is an endomorphism but not a ring automorphism. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to . This gives a mapping : Spec(Rp) Spec(R) of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field.
