On 2-superirreducible polynomials over finite fields

J.W. Bober, L. Du, D. Fretwell, G.S. Kopp, T.D. Wooley
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Abstract

We investigate -superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most . Let be a finite field of characteristic . We show that no 2-superirreducible polynomials exist in when and that no such polynomials of odd degree exist when is odd. We address the remaining case in which is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree . This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.
关于有限域上的 2 超可减多项式
我们研究-超可逆多项式,即在任何正阶数至多为.的多项式置换下仍然不可逆转的不可逆转多项式。设有限域的特征为 .我们证明,当 是奇数时,不存在 2 次超可重复性多项式;当 是奇数时,不存在奇数度的此类多项式。我们针对余下的情况,即奇数且多项式具有偶数阶,给出了具有偶数阶的单项式 2-superirreducible 多项式的明确公式.这个公式类似于高斯给出的有限域上给定阶数的一元不可还原多项式的个数公式。我们将讨论当多项式的度数或有限域的大小趋于无穷大时的相关渐近行为。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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