Acta Mathematica

Irreducibility of random polynomials of large degree

Emmanuel Breuillard and Péter P. Varjú

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We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.


E. B. acknowledges support from ERC Grant no. 617129 ‘GeTeMo’. P. V. acknowledges support from the Royal Society.

Article information

Acta Math., Volume 223, Number 2 (2019), 195-249.

Received: 31 October 2018
First available in Project Euclid: 16 April 2020

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11C08: Polynomials [See also 13F20]
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

random polynomials irreducibility Riemann hypothesis Dedekind zeta function Markov chains


Breuillard, Emmanuel; Varjú, Péter P. Irreducibility of random polynomials of large degree. Acta Math. 223 (2019), no. 2, 195--249. doi:10.4310/ACTA.2019.v223.n2.a1.

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