Hilbert's irreducibility theorem and the larger sieve

2010/11/30 by David Zywina, Zywina, David · 3 citations
Mathematics · #11F80 #11N36 (Secondary) #12E25 (Primary) 11G05 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1011.6465

openalex publication_date 2010/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe an explicit version of Hilbert's irreducibility theorem using a generalization of Gallagher's larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation associated to elliptic curves varying in rational families.

Cited by

• Larger sieve with height function and uniform bounds for integral points on curves over number fields
• Counting rational points on affine hypersurfaces
• Galois groups of random integer matrices

Related

• Hilbertian fields and Hilbert's irreducibility theorem
• On Hilbert's irreducibility theorem
• The principle of the large sieve
• The larger sieve and polynomial congruences
• Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants