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

 

Solution to a problem of S. Payne


Author: Xiang-dong Hou
Journal: Proc. Amer. Math. Soc. 132 (2004), 1-6
MSC (2000): Primary 11T06; Secondary 51E20
Published electronically: August 13, 2003
MathSciNet review: 2021242
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Abstract | References | Similar Articles | Additional Information

Abstract: A problem posed by S. Payne calls for determination of all linearized polynomials $f(x)\in\mathbb{F} _{2^n}[x]$ such that $f(x)$and $f(x)/x$ are permutations of $\mathbb{F} _{2^n}$ and $\mathbb{F} _{2^n}^*$respectively. We show that such polynomials are exactly of the form $f(x)=ax^{2^k}$ with $a\in\mathbb{F} _{2^n}^*$ and $(k,n)=1$. In fact, we solve a $q$-ary version of Payne's problem.


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Additional Information

Xiang-dong Hou
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
Address at time of publication: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: xhou@euler.math.wright.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07240-X
PII: S 0002-9939(03)07240-X
Keywords: Finite field, linearized polynomial, permutation polynomial
Received by editor(s): July 29, 2002
Published electronically: August 13, 2003
Additional Notes: This research was supported by NSA grant MDA 904-02-1-0080
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society