تقرير
On $2$-superirreducible polynomials over finite fields
| العنوان: | On $2$-superirreducible polynomials over finite fields |
|---|---|
| المؤلفون: | Bober, Jonathan W., Du, Lara, Fretwell, Dan, Kopp, Gene S., Wooley, Trevor D. |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11T06 (Primary), 12E05, 11S05 (Secondary) |
| الوصف: | 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 $\mathbb F$ be a finite field of characteristic $p$. We show that no $2$-superirreducible polynomials exist in $\mathbb F[t]$ 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. Comment: 10 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2309.15304 |
| رقم الأكسشن: | edsarx.2309.15304 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |