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Frobenius map (Definition)

Let $ K$ be any field of characteristic $ p > 0$, and suppose $ K$ contains the finite field $ \mathbb{F}_q$ of size $ q$, where $ q = p^r$. The $ q^{\rm th}$ power Frobenius map on $ K$ is the map $ \operatorname{Frob}_q: K \longrightarrow K$ defined by $ \operatorname{Frob}_q(x) := x^q$.

If $ K$ is perfect, then $ \operatorname{Frob}_q$ is an automorphism of $ K$ which fixes $ \mathbb{F}_q$, and accordingly is a member of the Galois group $ \operatorname{Gal}(K/\mathbb{F}_q)$.



"Frobenius map" is owned by djao.
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See Also: Frobenius homomorphism
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Cross-references: Galois group, automorphism, perfect, map, size, finite field, contains, characteristic, field
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This is version 3 of Frobenius map, born on 2002-04-14, modified 2002-12-02.
Object id is 2830, canonical name is FrobeniusMap.
Accessed 4263 times total.

Classification:
AMS MSC: 12E20 (Field theory and polynomials :: General field theory :: Finite fields )
 11T99 (Number theory :: Finite fields and commutative rings :: Miscellaneous)
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