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A generalization of Ore’s theorem on polynomials

  • Alexander V. Anashkin EMAIL logo
Published/Copyright: November 8, 2016
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 26 Issue 5

Abstract

Let GF(q) be the field of q elements and Vn(q) denote the n-dimensional vector space over the field GF(q). The linearized polynomial that corresponds to the polynomial f(x)=xn−∑i=0n−1cixi over the field GF(q) is the polynomial F(x)=xqn−∑i=0n−1cixqi. Let Tf denote the transformation of the vector space Vn(q) determined by the rule Tf(u0,...,un−2,un−1)=(u1,...,un−1,∑i=0n−1ciui). It is shown that if c0 ≠ 0, then the graph of the transformation Tf is isomorphic to the graph of the transformation Q: α → αq on the set of all roots of the polynomial F(x) in its splitting field. In this case the graph of the transformation Tf consists of cycles of lengths 1 ≤ d1 ≤ d2 ≤ ... ≤ dr if and only if the polynomial F(x) is the product of r + 1 irreducible factors of degrees 1, d1, d2, ... dr.

Keywords: linearized polynomial; primitive polynomial; isomorphism of graphs; Ore’s theorem

Originally published in Diskretnaya Matematika (2015) 27, №4, 21–25 (in Russian).

References

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[3] Nechaev A.A., Popov V.O., “A generalization of Ore’s theorem on irreducible polynomials over a fnite field”, Discrete Math. Appl, 25:4 (2015), 241-243.10.1515/dma-2015-0023Search in Google Scholar

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Received: 2015-4-27
Published Online: 2016-11-8
Published in Print: 2016-10-1

© 2016 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/dma-2016-0022
Keywords for this article
linearized polynomial; primitive polynomial; isomorphism of graphs; Ore’s theorem

Articles in the same Issue

  1. Frontmatter
  2. Research Article
  3. A generalization of Ore’s theorem on polynomials
  4. Research Article
  5. The sum of modules of Walsh coefficients of Boolean functions
  6. Research Article
  7. The algorithm for identical object searching with bounded worst-case complexity and linear memory
  8. Research Article
  9. Tests of contact closure for contact circuits
  10. Research Article
  11. Orbital derivatives over subgroups and their combinatorial and group-theoretic properties
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