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2017 Subgroup avoidance for primes dividing the values of a polynomial
Paul Pollack
Rocky Mountain J. Math. 47(6): 2043-2050 (2017). DOI: 10.1216/RMJ-2017-47-6-2043

Abstract

For $f \in \mathbb{Q} [x]$, we say that a rational prime~$p$ is a prime divisor of $f$ if $p$ divides the numerator of $f(n)$ for some integer $n$. Let $\mathcal{P} (f)$ denote the set of prime divisors of~$f$. We present an elementary proof of the following theo\-rem, which generalizes results of Bauer and Brauer: fix a nonzero integer~$g$. Suppose that $f(x) \in \mathbb{Q} [x]$ is a nonconstant polynomial having a root in $\mathbb{Q} _p$ for every prime $p$ dividing $g$, and having a root in $\mathbb{R} $ if $g \lt 0$. Let $m$ be a positive integer coprime to~$g$, and let~$H$ be a subgroup of $(\mathbb{Z} /m\mathbb{Z} )^{\times }$ not containing $g\bmod {m}$. Then there are infinitely many primes $p \in \mathcal{P} (f)$ with $p\bmod {m} \notin H$.

Citation

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Paul Pollack. "Subgroup avoidance for primes dividing the values of a polynomial." Rocky Mountain J. Math. 47 (6) 2043 - 2050, 2017. https://doi.org/10.1216/RMJ-2017-47-6-2043

Information

Published: 2017
First available in Project Euclid: 21 November 2017

zbMATH: 06816581
MathSciNet: MR3725255
Digital Object Identifier: 10.1216/RMJ-2017-47-6-2043

Subjects:
Primary: 11N32
Secondary: 11A41, 11N13

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 6 • 2017
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