Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

Abstract

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to $\F_p [x]$, for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1=2^{e_1}\cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0,...,f_s$ with $s\le e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.

10 pages. Fixed a typo in the references

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• Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms
• On the construction of irreducible polynomials over finite fields via prime degree endomorphisms of elliptic curves
• Sequences of irreducible polynomials without prescribed coefficients over finite fields of even characteristic