Abstract
Stickelberger–Swan Theorem is an important tool for determining parity of the number of irreducible factors of a given polynomial. Based on this theorem, we prove in this note that every affine polynomial A(x) over \({\mathbb{F}_2}\) with degree >1, where A(x) = L(x) + 1 and \({L(x)=\sum_{i=0}^{n}{x^{2^i}}}\) is a linearized polynomial over \({\mathbb{F}_2}\), is reducible except x 2 + x + 1 and x 4 + x + 1. We also give some explicit factors of some special affine pentanomials over \({\mathbb{F}_2}\) .
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Communicated by G. Mullen.
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Zhao, Z., Cao, X. A note on the reducibility of binary affine polynomials. Des. Codes Cryptogr. 57, 83–90 (2010). https://doi.org/10.1007/s10623-009-9350-7
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DOI: https://doi.org/10.1007/s10623-009-9350-7