Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.

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