Frobenius endomorphism

English Etymology Named after German mathematician Ferdinand Georg Frobenius. Noun en 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. en 2003 Claudia Miller The Frobenius endomorphism and homological dimensions Luchezar L. Avramov; Marc Chardin; Marcel Morales; Claudia Polini Commutative Algebra: Interactions with Algebraic Geometry: International Conference w:American Mathematical Society https://books.google.com.au/books?id=TnkbCAAAQBAJ&pg=PA208&dq=%22Frobenius+endomorphism%22%7C%22Frobenius+endomorphisms%22&hl=en&sa=X&ved=0ahUKEwjUuMLi9fveAhWfHjQIHb--DbgQ6AEISDAF#v=onepage&q=%22Frobenius%20endomorphism%22%7C%22Frobenius%20endomorphisms%22&f=false 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 , and let be a power of such that the group is defined over . We then denote by the corresponding Frobenius endomorphism. The Lie algebra and the adjoint action of on are also defined over and we still denote by the Frobenius endomorphism on . Assume that and the action of over are all defined over . Let and 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 . In this case, the characteristic polynomial of the Frobenius endomorphism denoted by (cf. Example 4.87 and Section 13.1.8), which sends to itself and to , is . Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points , we have . Synonyms particular endomorphism on a commutative ring with prime characteristic en Frobenius homomorphism Related terms en Frobenius automorphism en Frobenius closure en Frobenius element en Frobenius morphism Translations particular endomorphism on a commutative ring with prime characteristic French: fr endomorphisme de Frobenius m German: de Frobeniushomomorphismus m Italian: it endomorfismo di Frobenius m Polish: pl endomorfizm Frobeniusa m Further reading Characteristic (algebra) Frobenioid Perfect field Frobenius endomorphism on Encyclopedia of Mathematics Frobenius morphism on NLab nLab en eponyms