Math 2.0

Field theory and polynomials Definition

Frobenius automorphism of a finite field

Builds on Frobenius endomorphismFinite fieldRing isomorphism

Definition

Let FF be a finite field of characteristic pp. The Frobenius automorphism of FF is the map φ ⁣:F→F\varphi \colon F \to F, φ(x)=xp\varphi(x) = x^p.

Remarks

The map φ\varphi is the Frobenius endomorphism of FF. It is injective, as is every field homomorphism, so it is bijective, because FF is finite. The map φ\varphi fixes every element of the prime subfield 𝔽p\mathbb{F}_p, by Fermat's little theorem. If ∣F∣=pn|F| = p^n, then φn\varphi^n is the identity, and the automorphisms of FF are exactly id,φ,…,φn−1\mathrm{id}, \varphi, \dots, \varphi^{n-1}. So Gal(𝔽pn/𝔽p)\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p) is cyclic of order nn, generated by φ\varphi.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCharacteristic of a ring, definition, no evidence levelCharacteristic of a r…definitionCommutative ring, definition, no evidence levelCommutative ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionFreshman's dream, statement, no evidence levelFreshman's dreamstatementField, definition, no evidence levelFielddefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionFrobenius endomorphism, definition, no evidence levelFrobenius endomorphismdefinitionFinite field, definition, no evidence levelFinite fielddefinitionFrobenius automorphism of a finite field, definition, no evidence levelFrobenius automorphis…definition
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  1. Group concept
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  9. Ring isomorphism definition
  10. Frobenius endomorphism definition
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  12. Frobenius automorphism of a finite field definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Frobenius automorphism and Galois theory
  2. Web https://en.wikipedia.org/wiki/Frobenius_endomorphism, lead section
About this page
Tag0VP4TBVJ19
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Kinddefinition
WritingW0 stub
MSC12E20
DefinesFrobenius automorphism of a finite field
UsesFrobenius endomorphism, Finite field, Ring isomorphism

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