Math 2.0

Commutative algebra Definition

Frobenius endomorphism

Builds on Commutative ringCharacteristic of a ringFreshman's dreamRing homomorphism

Definition

Let pp be a prime number and let RR be a commutative ring of characteristic pp. The Frobenius endomorphism of RR is the map F ⁣:R→RF \colon R \to R, F(x)=xpF(x) = x^p. It is a ring homomorphism from RR to RR.

Remarks

The map FF preserves products because RR is commutative. It preserves sums by the freshman's dream. If RR is an integral domain, then FF is injective. If RR is a finite field, then FF is an automorphism of RR. On 𝔽p\mathbb{F}_p, the map FF is the identity, by Fermat's little theorem.

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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 homomorphismdefinitionFreshman's dream, statement, no evidence levelFreshman's dreamstatementFrobenius endomorphism, definition, no evidence levelFrobenius endomorphismdefinition
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  1. Group concept
  2. Ring definition
  3. Characteristic of a ring definition
  4. Commutative ring definition
  5. Ring homomorphism definition
  6. Freshman's dream statement
  7. Frobenius endomorphism definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Frobenius_endomorphism, Definition
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Kinddefinition
WritingW1 stated
MSC13A35
DefinesFrobenius endomorphism
UsesCommutative ring, Characteristic of a ring, Freshman's dream, Ring homomorphism

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