On irreducible polynomials of small height over finite fields
type: Item
instance of: scholarly article
title: On irreducible polynomials of small height over finite fields
zbMATH Open document ID: 0868.11056
DOI: 10.1007/BF01293260
author: Igor E. Shparlinski
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1997-08-19T00:00:00Z
review text: For a prime \(p\), let \(\mathbb{F}_p\) be the finite field of \(p\) elements and \(P_n(p)\) be the set of all monic irreducible polynomials of degree \(n\) over \(\mathbb{F}_p\). Also, let \(h_n(p)= \min_{f\in P_n(p)}H(f)\), where \(H(f)=\) greatest absolute value of the coefficients of \(f\) (here, for \(p\geq 3\), \(\mathbb{F}_p= \{0,\pm1,\dots, \pm\frac{p-1}{2}\}\)). The author proves the following result. For fixed \(n\geq 2\) and an arbitrary function \(q(x)\to\infty\), the bound \(h_n(p)\leq f(p)\) holds for all but \(o(H/\log X)\) primes \(p\) in the interval \([X-H,X]\), provided \(H\geq X^{7/12+ \varepsilon}\). The proof is based on some arguments which were used in proving Artin's conjecture on average.
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 938650
zbMATH Keywords: polynomials of small height
zbMATH Keywords: finite fields
reviewed by: A. Perelli
MaRDI profile type: Publication
cites work: Bombieri's theorem in short intervals. II
cites work: Q3717167
cites work: Q3137899
cites work: An average result for Artin's conjecture
cites work: Q3757982
full work available at URL: https://doi.org/10.1007/bf01293260
OpenAlex ID: W2057333655
description: scientific article; zbMATH DE number 938650