Reducibility of translates of Dickson polynomials
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- by Gerhard Turnwald PDF
- Proc. Amer. Math. Soc. 126 (1998), 965-971 Request permission
Abstract:
Let $K$ be a field and $a,b\in K$. The Dickson polynomial $D_{n}(x,a)$ is characterized by the equation $D_{n}(x+(a/x),a)=x^{n}+ (a/x)^{n}$. We prove that $D_{n}(x,a)+b\in K[x]$ is reducible if and only if there is a prime $p|n$ such that $b=-D_{p}(c,a^{n/p})$ for some $c\in K$, or $n=4k$ and $b=4c^{4}-8a^{k}c^{2}+2a^{2k}$ for some $c\in K$. This result generalizes the well-known reducibility criterion for binomials; and it provides a reducibility criterion for $T_{n}(x)+c$ where $T_{n}(x)$ denotes the Chebyshev polynomial of degree $n$.References
- W.-S. Chou: The factorization of Dickson polynomials over finite fields, Finite Fields Appl. 3 (1997), 84–96.
- Shuhong Gao and Gary L. Mullen, Dickson polynomials and irreducible polynomials over finite fields, J. Number Theory 49 (1994), no. 1, 118–132. MR 1295958, DOI 10.1006/jnth.1994.1086
- S. Lang: Algebra (Third Edition), Addison-Wesley, Reading, 1993.
- R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1237403
- Ladislaus Rédei, Algebra. Erster Teil, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 26, Teil 1, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1959 (German). MR 0106151
- Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
- Andrzej Schinzel, Selected topics on polynomials, University of Michigan Press, Ann Arbor, Mich., 1982. MR 649775, DOI 10.3998/mpub.9690541
Additional Information
- Gerhard Turnwald
- Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
- Email: gerhard.turnwald@uni-tuebingen.de
- Received by editor(s): September 10, 1996
- Communicated by: William W. Adams
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 965-971
- MSC (1991): Primary 12E10; Secondary 11T06
- DOI: https://doi.org/10.1090/S0002-9939-98-04363-9
- MathSciNet review: 1451832