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On the existence of some specific elements in finite fields of characteristic 2. (English) Zbl 1292.11136
Summary: Let $q$ be a power of 2, $n$ be a positive integer, and let $\Bbb F_{q^n}$ be the finite field with $q^n$ elements. In this paper, we consider the existence of some specific elements in $\Bbb F_{q^n}$. The main results obtained in this paper are listed as follows: (1) There is an element $\xi $ in $\Bbb F_{q^n}$ such that both $\xi $ and $\xi +\xi ^{-1}$ are primitive elements of $\Bbb F_{q^n}$ if $q=2^s$, and $n$ is an odd number no less than 13 and $s>4$. (2) For $q=2^s$, and any odd $n$, there is an element $\xi $ in $\Bbb F_{q^n}$ such that $\xi $ is a primitive normal element and $\xi +\xi^{-1}$ is a primitive element of $\Bbb F_{q^n}$ if either $n\mid (q-1)$, and $n\ge 33$, or $n\nmid (q-1)$, and $n\ge 30$, $s\ge 6$.

11T30Structure theory of finite fields
11T23Exponential sums
11T71Algebraic coding theory; cryptography
Full Text: DOI
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