Primitive element pairs with a prescribed trace in the quartic extension of a finite field. (English) Zbl 1479.12001
Summary: In this paper, we give a largely self-contained proof that the quartic extension \(\mathbb{F}_{q^4}\) of the finite field \(\mathbb{F}_q\) contains a primitive element \(\alpha\) such that the element \(\alpha + \alpha^{-1}\) is also a primitive element of \(\mathbb{F}_{q^4},\) and \(\mathrm{Tr}_{\mathbb{F}_{q^4} | \mathbb{F}_q} (\alpha)=a\) for any prescribed \(a\in \mathbb{F}_q\).
The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [the authors and R. K. Sharma, Finite Fields Appl. 54, 1–14 (2018; Zbl 1430.11158)].
The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [the authors and R. K. Sharma, Finite Fields Appl. 54, 1–14 (2018; Zbl 1430.11158)].
MSC:
| 12E20 | Finite fields (field-theoretic aspects) |
| 11T30 | Structure theory for finite fields and commutative rings (number-theoretic aspects) |
Citations:
Zbl 1430.11158References:
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| [2] | Cohen, S. D., Primitive elements and polynomials with arbitrary trace, Discr. Math.83 (1990) 1-7. · Zbl 0711.11048 |
| [3] | Gupta, A., Sharma, R. K. and Cohen, S. D., Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl.54 (2018) 1-14. · Zbl 1430.11158 |
| [4] | Robin, G., Estimation de la fonction de Tchebychef \(\theta\) sur \(k\)-ième nombre premier et grandes valeurs de la fonction \(\omega(n)\) nombre de diviseurs premiers de \(n\), Acta Arith.42 (1983) 367-389. · Zbl 0475.10034 |
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