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Frobenius map
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(Definition)
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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 $\Frob_q: K \longrightarrow K$ defined by $\Frob_q(x) := x^q$ .
If $K$ is perfect, then $\Frob_q$ is an automorphism of $K$ which fixes $\mathbb{F}_q$ , and accordingly is a member of the Galois group $\operatorname{Gal}(K/\mathbb{F}_q)$ .
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"Frobenius map" is owned by djao.
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Cross-references: Galois group, automorphism, perfect, map, size, finite field, contains, characteristic, field
There are 9 references to this entry.
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 5588 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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Pending Errata and Addenda
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