On 2-superirreducible polynomials over finite fields

Bober, Jonathan W. and Du, Lara and Fretwell, Dan and Kopp, Gene S. and Wooley, Trevor D. (2023) On 2-superirreducible polynomials over finite fields. Other. Arxiv.

[thumbnail of 2309.15304v1]
Text (2309.15304v1)
Download (0B)
[thumbnail of 2309.15304v1]
Text (2309.15304v1)
2309.15304v1.pdf

Download (149kB)

Abstract

We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. 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.

Item Type:
Monograph (Other)
Additional Information:
10 pages
Subjects:
?? math.nt11t06 (primary), 12e05, 11s05 (secondary) ??
ID Code:
220650
Deposited By:
Deposited On:
28 May 2024 15:25
Refereed?:
No
Published?:
Published
Last Modified:
28 May 2024 15:25