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Solving \(x+x^{2^l}+\ldots +x^{2^{ml}}=a\) over \(\mathbb{F}_{2^n} \). (English) Zbl 1446.11207
Summary: This paper presents an explicit representation for the solutions of the equation \(\sum_{i=0}^{\frac{k}{l}-1} x^{2^{li}} = a \in \mathbb{F}_{2^n}\) for any given positive integers \(k, l\) with \(l\mid k\) and \(n\), in the closed field \({\overline{\mathbb{F}_2}}\) and in the finite field \(\mathbb{F}_{2^n} \). As a by-product of our study, we are able to completely characterize the \(a\)’s for which this equation has solutions in \(\mathbb{F}_{2^n} \).

MSC:
11T06 Polynomials over finite fields
12E05 Polynomials in general fields (irreducibility, etc.)
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