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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Reducibility of translates
of Dickson polynomials


Author: Gerhard Turnwald
Journal: Proc. Amer. Math. Soc. 126 (1998), 965-971
MSC (1991): Primary 12E10; Secondary 11T06
MathSciNet review: 1451832
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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 $n$.


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Additional Information

Gerhard Turnwald
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: gerhard.turnwald@uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04363-9
PII: S 0002-9939(98)04363-9
Keywords: Dickson polynomials, Chebyshev polynomials, binomials, reducibility
Received by editor(s): September 10, 1996
Communicated by: William W. Adams
Article copyright: © Copyright 1998 American Mathematical Society



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