On $2$-superirreducible polynomials over finite fields
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914939159969792 |
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| author | Bober, Jonathan W. Du, Lara Fretwell, Dan Kopp, Gene S. Wooley, Trevor D. |
| author_facet | Bober, Jonathan W. Du, Lara Fretwell, Dan Kopp, Gene S. Wooley, Trevor D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15304 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $2$-superirreducible polynomials over finite fields Bober, Jonathan W. Du, Lara Fretwell, Dan Kopp, Gene S. Wooley, Trevor D. Number Theory 11T06 (Primary), 12E05, 11S05 (Secondary) 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. |
| title | On $2$-superirreducible polynomials over finite fields |
| topic | Number Theory 11T06 (Primary), 12E05, 11S05 (Secondary) |
| url | https://arxiv.org/abs/2309.15304 |