Abstract
We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).
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Notes
The code written in software Magma is shared in https://github.com/oguz-yayla/irredpoly. The amount of time for both methods is measured by using the online calculator http://magma.maths.usyd.edu.au/calc/ and hence they can be easily verified by the code.
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Acknowledgements
We would like to express our gratitude to the reviewers for their valuable comments, which improved the paper. Yağmur Çakıroğlu was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under the project number 116R026. Oğuz Yayla was supported by Middle East Technical University with the project number 10795. Emrah Sercan Yılmaz was supported by TÜBİTAK under the project number 117F274.
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Çakıroğlu, Y., Yayla, O. & Yılmaz, E.S. The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace. Des. Codes Cryptogr. 90, 2407–2417 (2022). https://doi.org/10.1007/s10623-022-01088-2
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DOI: https://doi.org/10.1007/s10623-022-01088-2