Frobenius endomorphism

plural Frobenius endomorphisms

1. commutative algebra; field theory; commutative algebra; field theoryalgebraGiven a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.

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

   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

   2 more examples

   2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11, Let k=◌̅ 𝔽_𝕡, and let q be a power of p such that the group G is defined over 𝔽_𝕢. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra 𝒢 and the adjoint action of G on 𝒢 are also defined over 𝔽_𝕢 and we still denote by F:𝒢→𝒢 the Frobenius endomorphism on 𝒢. […] Assume that H,X and the action of H over X are all defined over 𝔽_𝕢. Let F:X→X and F:H→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:y²+y=x³. In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ₂ (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is χ_E(T)=T²+2. Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( 𝔽_2ᵈ), we have ϕ=-[2]P.

   Synonyms: Frobenius homomorphism