Frobenius endomorphism

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English

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Etymology

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Named after German mathematician Ferdinand Georg Frobenius.

Noun

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Frobenius endomorphism (plural Frobenius endomorphisms)

  1. (algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps xx p for all xR.
    • 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 , 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 .

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Further reading

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