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The trace of 2-primitive elements of finite fields

Volume 192 / 2020

Stephen D. Cohen, Giorgos Kapetanakis Acta Arithmetica 192 (2020), 397-419 MSC: Primary 11T30; Secondary 11T06. DOI: 10.4064/aa190307-23-5 Published online: 29 November 2019


Let $q$ be a prime power and $n, r$ integers such that $r\,|\, q^n-1$. An element of $\mathbb F _{q^n}$ of multiplicative order $(q^n-1)/r$ is called $r$-primitive. For any odd prime power $q$, we show that there exists a $2$-primitive element of $\mathbb F _{q^n}$ with arbitrarily prescribed $\mathbb F _q$-trace when $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$.


  • Stephen D. Cohen6 Bracken Road
    Aberdeen AB12 4TA, Scotland, UK
  • Giorgos KapetanakisDepartment of Mathematics
    and Applied Mathematics
    University of Crete
    Voutes Campus
    70013 Heraklion, Greece

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