PlanetMath: Frobenius homomorphism

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Frobenius homomorphism

(Definition)

Let be a field of characteristic . Then for any $a, b \in F$ ,

$\displaystyle (a + b)^p$	$\displaystyle =$	$\displaystyle a^p + b^p,$
$\displaystyle (ab)^p$	$\displaystyle =$	$\displaystyle a^p b^p.$

Thus the map

$\displaystyle \begin{matrix}\phi: F &\to& F \\ a &\mapsto& a^p \end{matrix}$

is a field homomorphism, called the Frobenius homomorphism, or simply the Frobenius map on

. If it is surjective then it is an automorphism, and is called the Frobenius automorphism.

Note: This morphism is sometimes also called the “small Frobenius” to distinguish it from the map $a \mapsto a^q$ , with . This map is then also referred to as the “big Frobenius” or the “power Frobenius map”.

"Frobenius homomorphism" is owned by mathcam. [ full author list (5) | owner history (3) ]

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See Also: Frobenius morphism, Frobenius map

Other names:

Frobenius endomorphism, Frobenius map

Also defines:

Frobenius automorphism

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Cross-references: morphism, automorphism, surjective, field homomorphism, map, characteristic, field
There are 4 references to this entry.

This is version 8 of Frobenius homomorphism, born on 2002-02-18, modified 2006-10-04.
Object id is 2157, canonical name is FrobeniusAutomorphism.
Accessed 5463 times total.

Classification:

AMS MSC:

12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 3 ]

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