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What (who) is arithmetic and logic unit - definition
![The [[combinational logic]] circuitry of the [[74181]] integrated circuit, an early four-bit ALU](https://commons.wikimedia.org/wiki/Special:FilePath/74181aluschematic.png?width=200 "The [[combinational logic]] circuitry of the [[74181]] integrated circuit, an early four-bit ALU")
![The [[hyperelliptic curve]] defined by <math>y^2=x(x+1)(x-3)(x+2)(x-2)</math> has only finitely many [[rational point]]s (such as the points <math>(-2, 0)</math> and <math>(-1, 0)</math>) by [[Faltings's theorem]].](https://commons.wikimedia.org/wiki/Special:FilePath/Example of a hyperelliptic curve.svg?width=200 "The [[hyperelliptic curve]] defined by <math>y^2=x(x+1)(x-3)(x+2)(x-2)</math> has only finitely many [[rational point]]s (such as the points <math>(-2, 0)</math> and <math>(-1, 0)</math>) by [[Faltings's theorem]].")
Wikipedia
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 φ that takes r in R to rp is a ring endomorphism of R.
The image of φ is then Rp, the subring of R consisting of p-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring automorphism.
The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping
- φ*: Spec(Rp) → Spec(R)
of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field.
Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.
