Turkish Journal of Mathematics
DOI
10.55730/1300-0098.3413
Abstract
Let $\mathbb{F}_q[x]$ be the ring of polynomials over a finite field $\mathbb{F}_q$ and $\mathbb{F}_q(x)$ its quotient field. Let $\mathbb{P}$ be the set of primes in $\mathbb{F}_q[x]$, and let $\mathcal{I}$ be the set of all polynomials $f$ over $\mathbb{F}_q(x)$ for which $f(\mathbb{P})\subseteq\mathbb{F}_q[x]$. The existence of a basis for $\mathcal{I}$ is established using the notion of characteristic ideal; this shows that $\mathcal{I}$ is a free $\mathbb{F}_q[x]$-module. Through localization, explicit shapes of certain bases for the localization of $\mathcal{I}$ are derived, and a well-known procedure is described as to how to obtain explicit forms of some bases of $\mathcal{I}$.
Keywords
Integer-valued polynomial, polynomial over finite field, prime, localization
First Page
1073
Last Page
1086
Recommended Citation
CHAICHANA, TUANGRAT; LAOHAKOSOL, VICHIAN; MEESA, RATTIYA; and YUTTANAN, BOONROD
(2023)
"Polynomials taking integer values on primes in a function field,"
Turkish Journal of Mathematics: Vol. 47:
No.
4, Article 3.
https://doi.org/10.55730/1300-0098.3413
Available at:
https://journals.tubitak.gov.tr/math/vol47/iss4/3
