Frobenius endomorphism

The map on a field of characteristic p that sends each element x to x^p. It preserves addition and multiplication, and is injective; it is surjective when the field is perfect.

Characteristic of a field: The least positive integer n such that adding the multiplicative identity n times gives zero, or zero if none exists. Characteristic p makes the binomial coefficients between the endpoints vanish modulo p.
Finite field arithmetic: The algebra of computations in fields with finitely many elements. Repeated Frobenius powers provide efficient identities and structure in finite-field calculations.
Field (mathematics): A commutative ring in which every nonzero element has a multiplicative inverse. Frobenius is defined on fields, where its preservation of operations makes it an endomorphism.
Identity function: The function that maps every element of a set to itself. Frobenius fixes the prime subfield but generally moves other field elements.
Freshman's dream: The identity (a+b)^p = a^p+b^p in a commutative ring of characteristic p. This identity makes the pth-power map preserve addition.
Frobenius automorphism: An automorphism of a field that acts by raising elements to a prime-power exponent. For finite fields, the pth-power map is the basic Frobenius automorphism.
Prime number: An integer greater than one with no positive divisors except one and itself. The exponent p is the prime characteristic of the field.
Field automorphism: A bijective homomorphism from a field to itself. Frobenius is an automorphism only when it is surjective, as for perfect fields.
Prime subfield: The smallest subfield of a field, generated by its multiplicative identity. Every element of the prime subfield is fixed by the pth-power map.
Galois group of a finite field: The cyclic group of automorphisms of a finite field extension, generated by a Frobenius map. Its generator is the pth-power map over the prime field.

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