Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Reducibility of translates of Dickson polynomials
HTML articles powered by AMS MathViewer

by Gerhard Turnwald PDF
Proc. Amer. Math. Soc. 126 (1998), 965-971 Request permission


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 $n$.
  • 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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 12E10, 11T06
  • Retrieve articles in all journals with MSC (1991): 12E10, 11T06
Additional Information
  • Gerhard Turnwald
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
  • Email:
  • 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:
  • MathSciNet review: 1451832