Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class that includes finite fields.

for q < p; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients if 1 ≤ k ≤ p − 1.

If φ : R → S is a homomorphism of rings of characteristic p, then If FR and FS are the Frobenius endomorphisms of R and S, then this can be rewritten as: This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic p rings to itself.

Then the image of F does not contain t. If it did, then there would be a rational function q(t)/r(t) whose p-th power q(t)p/r(t)p would equal t. But the degree of this p-th power (the difference between the degrees of its numerator and denominator) is p deg(q) − p deg(r), which is a multiple of p. In particular, it can't be 1, which is the degree of t. This is a contradiction; so t is not in the image of F. A field K is called perfect if either it is of characteristic zero or it is of positive characteristic and its Frobenius endomorphism is an automorphism.

If R is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field.

-algebra, then the fixed points of the nth iterate of Frobenius are the elements of the image of

The Galois group of an extension of finite fields is generated by an iterate of the Frobenius automorphism.

The Frobenius automorphism F of Fq fixes the prime field Fp, so it is an element of the Galois group Gal(Fq/Fp).

If n > 1, then the Frobenius automorphism F of Fqf does not fix the ground field Fq, but its nth iterate Fn does.

The Galois group Gal(Fqf /Fq) is cyclic of order f and is generated by Fn.

The generators of Gal(Fqf /Fq) are the powers Fni where i is coprime to f. The Frobenius automorphism is not a generator of the absolute Galois group because this Galois group is isomorphic to the profinite integers which are not cyclic.

However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme.

This endomorphism is called the absolute Frobenius morphism of X, denoted FX.

The absolute Frobenius morphism is a natural transformation from the identity functor on the category of Fp-schemes to itself.

Composing φ with FS results in an S-scheme XF called the restriction of scalars by Frobenius.

For example, consider a ring A of characteristic p > 0 and a finitely presented algebra over A: The action of A on R is given by: where α is a multi-index.

The extension of scalars by Frobenius is defined to be: The projection onto the S factor makes X(p) an S-scheme.

As before, consider a ring A and a finitely presented algebra R over A, and again let X = Spec R. Then: A global section of X(p) is of the form: where α is a multi-index and every aiα and bi is an element of A.

The action of an element c of A on this section is: Consequently, X(p) is isomorphic to: where, if: then: A similar description holds for arbitrary A-algebras R. Because extension of scalars is base change, it preserves limits and coproducts.

This implies in particular that if X has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does X(p).

The arithmetic and geometric Frobenius morphisms are then endomorphisms of X, and so they lead to an action of the Galois group of k on X.

This set comes with a Galois action: Each such point x corresponds to a homomorphism OX → K from the structure sheaf to K, which factors via k(x), the residue field at x, and the action of Frobenius on x is the application of the Frobenius morphism to the residue field.

Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of X.

Given an unramified finite extension L/K of local fields, there is a concept of Frobenius endomorphism that induces the Frobenius endomorphism in the corresponding extension of residue fields.

We may define the Frobenius map for elements of the ring of integers OL of L as an automorphism sΦ of L such that 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.

We may find the image of ρ under the Frobenius map by locating the root nearest to ρ3, which we may do by Newton's method.

However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if p-adic results will suffice.

If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime φ in the base field K. For an example, consider the extension Q(β) of Q obtained by adjoining a root β satisfying to Q.

This extension is cyclic of order five, with roots for integer n. It has roots that are Chebyshev polynomials of β: give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form 22n + 1 (which split).

It is immediately apparent how the Frobenius map gives a result equal mod p to the p-th power of the root β.