On the number of times a root of \(f(n,x)=0\) generates a field containing a given number field
type: Item
instance of: scholarly article
title: On the number of times a root of \(f(n,x)=0\) generates a field containing a given number field
zbMATH Open document ID: 0928.11043
published in: Journal of Number Theory
publication date: 1999-11-10T00:00:00Z
review text: Let \(f(x,y)\) be a polynomial with rational coefficients. The author obtains upper bounds for the number \(N(T)\) of integers \(n\leq T\) such that there exists a root of \(f(n,y)=0\) generating a field which contains a given algebraic number field \(E\). In the case when \(f\) is irreducible over the rationals and monic in \(y\) and \(E\) is not contained in the algebraic closure \({\mathcal C}\) of the rationals in the splitting field \(\Lambda\) of \(f\) over \(\mathbb{Q}(x)\), then \(N(T)\) does not exceed \(c^{\log T/\log\log T}\sqrt T\) with \(c\) being dependent only on \(f\) but not on \(E\). If moreover every intermediate field \({\mathcal C}(x)\subset\Omega\subset\Lambda\) (\(\Omega\neq{\mathcal C}(x)\)) has positive genus and is ramified above at least one irrational point then the factor \(\sqrt T\) in this bound can be omitted. As an application one obtains a uniform version of Hilbert's irreducibility theorem for number fields.
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12E25
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 1229751
zbMATH Keywords: Hilbert's irreducibility theorem
zbMATH Keywords: field extensions
zbMATH Keywords: polynomials
author: Umberto Zannier
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: jnth.1998.2243
OpenAlex ID: W2006160689
cites work: Q3209230
cites work: Squares in arithmetic progressions
cites work: The Distribution of Galois Groups and Hilbert's Irreducibility Theorem
cites work: Q4319817
cites work: Q3309989
cites work: On the prime ideal theorem and irregularities in the distribution of primes
cites work: Diophantine approximations and diophantine equations
cites work: Q4233886
DOI: 10.1006/JNTH.1998.2243
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description: scientific article; zbMATH DE number 1229751