## Solution to a problem of S. Payne

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- by Xiang-dong Hou PDF
- Proc. Amer. Math. Soc.
**132**(2004), 1-6 Request permission

## 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.## References

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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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 1-6 - MSC (2000): Primary 11T06; Secondary 51E20
- DOI: https://doi.org/10.1090/S0002-9939-03-07240-X
- MathSciNet review: 2021242