Frobenius endomorphism

English

Etymology

Named after German mathematician Ferdinand Georg Frobenius.

Noun

Frobenius endomorphism (plural Frobenius endomorphisms)

1. (algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → x ^p for all x ∈ R.
   • 2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 208:

     Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.

   • 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,

     Let $k={\overline {\mathbb {F} }}_{p}$, and let $q$ be a power of $p$ such that the group $G$ is defined over $\mathbb {F} _{q}$. We then denote by $F:G\to G$ the corresponding Frobenius endomorphism. The Lie algebra ${\mathcal {G}}$ and the adjoint action of $G$ on ${\mathcal {G}}$ are also defined over $\mathbb {F} _{q}$ and we still denote by $F:{\mathcal {G}}\to {\mathcal {G}}$ the Frobenius endomorphism on ${\mathcal {G}}$.

     […] Assume that $H,X$ and the action of $H$ over $X$ are all defined over $\mathbb {F} _{q}$. Let $F:X\to X$ and $F:H\to H$ be the corresponding Frobenius endomorphisms.

   • 2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,

     The first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve

     $E:y^{2}+y=x^{3}$.

     In this case, the characteristic polynomial of the Frobenius endomorphism denoted by $\phi _{2}$ (cf. Example 4.87 and Section 13.1.8), which sends $P_{\infty }$ to itself and $(x_{1},y_{1})$ to $(x_{1}^{2},y_{1}^{2})$, is

     $\chi _{E}(T)=T^{2}+2$.

     Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points $P\in E(\mathbb {F} _{2^{d}})$, we have $\phi _{2}^{2}=-[2]P$.

Synonyms

• (particular endomorphism on a commutative ring with prime characteristic): Frobenius homomorphism

Related terms

• Frobenius automorphism
• Frobenius closure
• Frobenius element
• Frobenius morphism

Translations

particular endomorphism on a commutative ring with prime characteristic

• French: endomorphisme de Frobenius m
• German: Frobeniushomomorphismus m
• Italian: endomorfismo di Frobenius m

Further reading

• Characteristic (algebra) on Wikipedia.Wikipedia
• Frobenioid on Wikipedia.Wikipedia
• Perfect field on Wikipedia.Wikipedia
• Frobenius endomorphism on Encyclopedia of Mathematics
• Frobenius morphism on nLab