Normal Elements and Irreducible Polynomials in Finite Fields

Wei CAO, Wei Hua LI, Bi Yun XU

Acta Mathematica Sinica, Chinese Series ›› 2024, Vol. 67 ›› Issue (4) : 624-633.

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PDF(477 KB)
Acta Mathematica Sinica, Chinese Series ›› 2024, Vol. 67 ›› Issue (4) : 624-633. DOI: 10.12386/A20220014

Normal Elements and Irreducible Polynomials in Finite Fields

  • Wei CAO1,2, Wei Hua LI3, Bi Yun XU3
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Abstract

Let Fq be the finite field of q elements, and Fqn be its extension of degree n. An element αFqn is called a normal element of Fqn/Fq if {α,αq,,αqn1} constitutes a basis of Fqn/Fq. Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace, while the converse does not hold in general. Using linearized polynomials, we give some necessary and sufficient conditions for this problem, which extend the known results.

Key words

finite field / normal element / linearized polynomial / irreducible polynomial

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Wei CAO, Wei Hua LI, Bi Yun XU. Normal Elements and Irreducible Polynomials in Finite Fields. Acta Mathematica Sinica, Chinese Series, 2024, 67(4): 624-633 https://doi.org/10.12386/A20220014

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